Some results of convex programming complexity

Size: px
Start display at page:

Download "Some results of convex programming complexity"

Transcription

1 2012c12 $ Ê Æ Æ 116ò 14Ï Dec., 2012 Oerations Research Transactions Vol.16 No.4 Some results of convex rogramming comlexity LOU Ye 1,2 GAO Yuetian 1 Abstract Recently a number of aers were written that resent low-comlexity interior-oint methods for different classes of convex rograms. To guarantee the olynomiality of the rocedure, in this aer we show that the logarithmic barrier function associated with these rograms is self-concordant. In other words, we will resent two different lemmas with different logarithmic barrier functions and aly them to several classes of structured convex otimization roblems, using the self-concordancy. Keywords convex rogramming, self-concordant barrier functions, entroy rogramming, otimization comlexity Chinese Library Classification O Mathematics Subject Classification 90C25 'u à5y KE,5ïÄ(J ð 1,2 U 1 Á 8c ulœþïäˆaøóà5y$e,ýæn¼ê { Ù. ^ gúnø éøóaaà5y KEƒAéêæN¼ê ÏLü Úny²ùà 5y KƒAéêæN¼êÑ vgú ŠâNesterov ÚNemirovskyóŠy² KS:Ž{äkõ ªE,5. ' c à5y gúæn¼ê 5y `ze,5 ã aò O221.2 êæ aò(2010) 90C25 0 Introduction Convex otimization deals with the following roblem (CL) inf x R n f 0 (x), s.t. x C, where C R n is a closed convex set and f 0 : C R is a convex function defined on C. ÂvFϵ2012c1120F 1. Deartment of Mathematics, Shanghai University, Shanghai , China; þ ŒÆêÆX, þ, þ ÆEâ Æ þ, ; Shanghai Vocational College of Science and Technology, Shanghai , China ÏÕŠö Corresonding author

2 4Ï Some results of convex rogramming comlexity 113 A fundamental ingredient in the elaboration of these methods is barrier method. Namely, consider the following arameterized family of unconstrained minimization roblem: (CL µ ) inf x R n f 0 (x) µ + F (x), where arameter µ belongs to R ++ and is called the barrier arameter. The constraint x C of the original roblem (CL) has been relaced by a enalty term F (x) in the objective function, which tends to + as x tends to the boundary of C and whose urose is to avoid that the iterates leave the feasible set. Assuming existence of a minimizer x(µ) for each of these roblems(strong convexity of F ensures uniqueness of such a minimizer x(µ)), the set {x(µ) µ > 0} C is called the central ath for the roblem (CL). It is intuitively clear that as µ tends to zero, the first term roortional to the original objective f0(x) µ becomes reonderant in the sum, which imlies that the central ath converges to a solution that is otimal for the original roblem. The rincile behind interioroint methods will thus be to follow this central ath until an iterate that is sufficiently close to the otimum is found. The efficiency of a barrier method for solving convex rograms strongly deends on the roerties of the barrier function used. A key roerty that is sufficient to rove olynomial convergence for barrier methods is the roerty of self-concordance introduced in [1]. This condition not only allows a roof of olynomial convergence, but numerical exeriments in [2-3] and others further indicate that numerical algorithms based on self-concordant barrier functions are of ractical interest and effectively exloit the structure of the underlying roblems. In this article we resent two different lemmas, if the barrier function satisfies the condition of corresonding lemma, the function is also self-concordant. We will show that for several classes of convex roblems for which interior-oint methods were resented in the literature the logarithmic barrier function is self-concordant. 1 Some general comosition rules Firstly we give a recise definition of self-concordance as given by [1]. Definition 1.1 A function F : C R is called (κ, ν)-self-concordant for the convex set C R n if and only if F is a barrier function and the following two conditions hold for all x intc and h R n : 3 F (x)[h, h, h] 2κ( 2 F (x)[h, h]) 3 2, (1.1) F (x) T ( 2 F (x)) 1 F (x) ν (1.2) (Note that the square root is well defined since its argument 2 F (x)[h, h] is ositive of the requirement that F is convex). Furthermore, we call a barrier function F (x) to be ν-logarithmically homogeneous if F (tx) = F (x) ν log t

3 114 LOU Ye, GAO Yuetian 16ò for all x F 0 and t > 0. Just as the definition of ordinary convexity, self-concordancy is a line-roerty, i.e., the definition of a self-concordant function can be restricted to any line lying in the domain. To see this, let Then, d(t) := F (x + th). d (1) (0) = F (x)[h], d (2) (0) = 2 F (x)[h, h], d (3) (0) = 3 F (x)[h, h, h]. Therefore, F (x) is a self-concordant function satisfying 3 F (x)[h, h, h] 2κ(h T 2 F (x)h) 3/2. if and only if it is a self-concordant function restricted to any line in its domain, i.e. d (3) (0) 2κ(d (2) (0)) 3/2, for any given x in its domain and any given feasible direction h. This observation allows us to rove the self-concordant roerty of a function by roving this roerty for the function restricted to an arbitrary line in its domain. We are now in osition to sketch a short-ste algorithm. Given a roblem of tye (CL), a barrier function F for C, an uer bound on the roximity measure τ > 0, a decrease arameter 0 < θ < 1 and an initial iterate x 0 such that δ(x 0, µ 0 ) < τ, we set k 0 and erform the following main loo: (1) µ k+1 µ k (1 θ); (2) x k+1 x k + n µk+1 (x k ); (3) k k + 1. The key is to choose arameters τ and θ such that δ(x k, µ k ) < τ imlies δ(x k+1, µ k+1 ) < τ in order to reserve roximity to the central ath. This crucial question is answered by the remarkable theory of self-concordant functions. The following theorem gives the final comlexity results in [11]. Theorem 1.1 Given a convex otimization roblem (CL), a (κ, ν)-self-concordant barrier F for C and an initial iterate x 0 such that δ(x 0, µ 0 ) < κ, one can find a solution with accuracy ε in ( κ ν) log 1.29µ 0κ ν. ε This theorem imlies that with the hel of self-concordancy of the barrier function F, short-ste interior-oint methods is low-comlexity. The next lemma gives some helful comosition rules for self-concordant functions. The roof follows immediately from the definition of self-concordance. Lemma 1.1 (Nesterov and Nemirovsky [1] )

4 4Ï Some results of convex rogramming comlexity 115 (1) (addition and scaling) Let ϕ i be κ i -self-concordant on F 0 i, i = 1, 2, and ρ 1, ρ 2 R + ; then ρ 1 ϕ 1 + ρ 2 ϕ 2 is κ-self-concordant on F 0 1 F 0 2, where κ = max{κ 1 / ρ 1, κ 2 / ρ 2 }. (2) (affine invariance) Let ϕ be κ-self-concordant on F 0 and let B(x) = Bx + b : R k R n be an affine maing such that B(R k ) F 0. Then ϕ(b( )) is κ-self-concordant on {x : B(x) F 0 }. 2 Alication I Now consider a standard convex rogramming roblem (CP) min c T x s.t. Ax = b, f i (x) 0, i = 1,, m, where f i (x) is smooth and convex, i = 1,, m. For simlicity, let m = 1 and f(x) = f 1 (x). Also let the decision variable now be and the roblem data as 0 c := 0 c x := q x R 1 R 1 R n, R 1 R 1 R n, b := 1 0 b R 1 R 1 R m (2.1) and Ā := T T 0 0 A R (m+2) (n+2). (2.2) Let K = cl{ x > 0, q f(x/) 0} R n+2, which is a closed cone. Ye [12] has roved that it is also convex. An equivalent formulation for (CP) is given by (CCP) min s.t. c T x Ā x = b, x K. Naturally, a 2-logarithmically homogeneous and convex barrier function for K is F ( x) = log log(q f(x/)).

5 116 LOU Ye, GAO Yuetian 16ò u It holds that K = cl s = v v > 0, u vf (s/v) 0 s and F ( s) = log v log(u vf (s/v)) is a 2-logarithmically homogeneous barrier function for K. The result above was roved by Zhang [13]. The function in this article is of the tye F ( x) = log log(q f(x/)). Now let us consider the following function in R 1 F () = log log g(), where g() = q f(x/) 0. Simly calculation shows that F (1) () = g(1) () g() F (2) () = g(2) () g() F (3) () = g(3) () g() 1, (2.3) + [ g (1) ] 2 () + 1 g() + 3g(2) ()g (1) () (g()) 2 2 2, (2.4) [ g (1) ] 3 () 2 g() 3. (2.5) The next lemma gives a sufficient condition for an objective function F to guarantee that combined with the logarithmic barrier function for the ositive orthant R n + of R is self-concordant. This lemma will hel to simlify self-concordance roofs in the sequel. Lemma 2.1 If there exists a k > 0 such that g() satisfies > 0, then F () is self-concordant. g (3) () k g(2) (), (2.6) Proof In that case g() = q f(x/) 0( > 0), because f is convex, we can derive that g (2) () = 1 3 x T 2 f(x/)x 0. The three terms on the right-hand side of (1.4) are nonnegative, i.e., the right-hand side can be abbreviated by F (2) () = a 2 + b 2 + c 2, with a 2 = g(2) () g(), b2 = [ ] g (1) 2 () g(), c 2 = 1. Obviously, 2 a F (2) () 1/2, b F (2) () 1/2, c F (2) () 1/2.

6 4Ï Some results of convex rogramming comlexity 117 So we can bound the right-hand side of (1.5) by F (3) () g (3) () g() + 3g (2) ()g (1) () [ (g()) 2 + g (1) ] 3 2 () g() k g (2) () g() + 3 g (2) () g (2) () g() g() + 2 g (2) 2 () g() = k c a 2 + 3a 2 b + 2 b c 3 k F (2) () 3/2 + 3 F (2) () 3/2 + 2 F (2) () 3/2 + 2 F (2) () 3/2 = (7 + k) F (2) () 3/2. In articular, if f is a convex quadratic function, of course it satisfies g (3) () k g(2) (), F ( x) is self-concordant roved by Zhang in [13]. Now we consider the general formulation of (CP) where m 1. Similarly we have its reresentation (CCP) with m K = K i, where K i = cl{ x > 0, q f i (x/) 0} R n+2, i = 1,, m. The natural 2m-logarithmically homogeneous barrier function for K is F ( x) = m log m log(q f i (x/)). 3 The dual cone of K is m m K = cl(k1 Km) = cl s i = u i v i s i v i > 0, u i v i fi ( si v i ) 0, i = 1,, m and the dual barrier function for K is also 2m-logarithmically homogeneous, is given as follows m F ( s 1,, s m ) = [log v i + log(u i v i fi (s i /v i ))]. Extended entroy otimization The extended entroy rogramming roblem is defined as (EEP) min c T x + s.t. Ax = b, x 0. g i (x i ) Where A is an m n matrix and c and b are n-and m-dimensional vectors, resectively. Moreover, it is assumed that the scalar functions g i C 3 satisfy g (3) i (x i ) k i g (2) i (x i )/x i, i = 1,, n. In the case of entroy rogramming we have g i (x i ) = x i ln x i, for all i, and k i = 1.

7 118 LOU Ye, GAO Yuetian 16ò An equivalent formulation for (EEP) is given by min Z s.t. c T x + Ax = b, x 0. g i (x i ) Z 0, Consequently, we can get another equivalent formulation (CEEP) for (EEP) (CEEP) min s.t. Z Ā x = b, { ( x K = cl x c T > 0, q x n + ( xi ) ) } g i + Z 0, x 0. Where f(x) = c T x + n g i (x i ) Z and Ā, x, b are defined as (1.1), (1.2). Lemma 2.2 Suose that g (3) i (x i ) k i g (2) i (x i )/x i, i = 1,, n, let x = q x R n+2, then the logarithmic barrier function F ( x) = log(q f(x/)) log for the extended entroy rogramming roblem (CEEP) is self-concordant in the domain > 0 and q f(x/) > 0 where f(x) = c T x + n g i (x i ) Z. Proof Consider an arbitrary line in the domain of F ( x). If the comonent remains constant along this line, then by Lemma 1.2 the function F ( x) is self-concordant on the line. Let us consider the case where changes along the line. We may assume that the line is arameterized by, i.e. q = a 0 + b 0, x i = a i + b i (i = 1,, n), Z = a n+1 + b n+1 characterize the line, where serves as the arameter. We have that ( c T x g() = q = a 0 + b 0 + n ( xi ) ) g i + Z ( ai ) c i (a i + b i ) g i + b i + a n+1 + b n+1.

8 4Ï Some results of convex rogramming comlexity 119 Simly calculation shows that g (1) () = b 0 + b n+1 c i b i + [ ( ai ) g i + b i a ( i ai ) ] g(1) i + b i, g (2) a 2 ( i ai ) () = 3 g(2) i + b i, [ 3a g (3) 2 ( () = i ai ) ( 4 g(2) i + b i + a3 i ai ) ] 5 g(3) i + b i. Let u i = ai + b i > 0, we have ai = u i b i. Obviously the extended entroy rogramming roblem satisfies g (3) i (x i ) k i g (2) i (x i )/x i, i = 1,, n. Consequently, [ g (3) 3a 2 i () 4 g(2) i ( a i + b i) + a 3 i 5 g(3) i ( a ] i + b i) [ 3a 2 i = 4 g(2) i (u i ) + a 3 ] i 5 g(3) i (u i ) [ ] 3a 2 i 4 g(2) i (u i ) + k a 2 i i 4 (u i b i ) g(2) i (u i ) 1 ( 3 + k i(1 b ) i ) a 2 i u i 3 g(2) i (u i ), let k = max k i, M = max 1 bi u i, i = 1,...n, then the above exression can be abbreviated by g (3) () 3 + km a 2 i 3 g(2) i (u) = 3 + km g (2) (). Follow the Lemma 1.2, naturally F (3) (x) [7 + (3 + km)] F (2) (x) 3/2 = (10 + km) F (2) (x) 3/2. This shows that F ( x) is self-concordant in all directions within its domain. The desired result is roved. Similarly, we can use the result of Lemma 1.2 in dual geometric rogramming roblem, Monteiro and Adler s condition and other smoothes condition in [14]. Theorem 2.1 Consider convex otimization roblem (CP) where f i (x) are all convex and g i () = q f i (x/) satisfies the condition g (2) g i () k (2) i () i, i = 1,, n, then the barrier function for the formulation (CCP) F ( x) = n log log(q f i (x/)) is self-concordant with a comlexity value in the order of n. u i

9 120 LOU Ye, GAO Yuetian 16ò 3 Alication II This lemma we are going to resent deals with the first-concordancy condition. Let us first introduce two auxiliary function r 1 and r 2 : { } { } γ γ /γ r 1 : R R : γ max 1, and r 2 : R R : γ max 1,. 3 2/γ 3 + 4/γ + 2/γ 2 Both of these functions are equal to 1 for γ 1 and strictly increasing for γ 1, with the asymtotic aroximations r 1 (γ) γ 3 and r 2 (γ) γ+1 3 when γ tends to +. Lemma 3.1 Let us suose F is a convex function with effective domain C R n + and that there exists a constant γ such that 3 F (x)[h, h, h] 3γ 2 F (x)[h, h] n h 2 i for all x intc, h R n. (3.1) We have that x 2 i F 1 : C R : x F (x) log x i satisfies the first condition of self-concordancy (1.2) with arameter κ 1 = r 1 (γ) on its domain C and F 2 : C R R : (x, u) log(u F (x)) log x i satisfies the first condition of self-concordancy (1.2) with arameter κ 2 = r 2 (γ) on its domain ei F = {(x, u) F (x) u} This lemma is roved in [11]. While the revious lemma is use to tackle the first condition of self-concordant (1.1), it does not say anything about of the second condition (1.2). The following corollary about the second barrier F 2 might rove useful in the resect. Corollary 3.1 Let F satisfy the assumtion of Lemma 3.1. Then the second barrier is (r 2 (γ), n + 1)-self-concordant. F 2 : C R R : (x, u) log(u F (x)) log x i Glineur in [11] has used the results of Lemma 3.1 and Corollary 3.2 to several classes of structured convex otimization roblems, such as extended entroy otimization, dual geometric otimization and l -norm otimization, which are shown to admit a self-concordant logarithmic barrier. In this article we will extend the alication to other convex rogramming. The dual l -rogramming roblem

10 4Ï Some results of convex rogramming comlexity 121 Let q i be such that 1/ i + 1/q i = 1, 1 i m, and let the rows of a matrix A be a i, i = 1,, m, and the rows of a matrix B be b k, k = 1,, r. Then, the dual of the l -rogramming roblem (PL ) is (DL ) inf c T y + d T z + A T y + B T z = η, z 0 r z k k=1 i I k (1/q i ) y i /z k qi, (If y i 0 and z k = 0, then z k y i /z k qi is defined as ). The above roblem is equivalent to inf c T y + d T z + s qi i z qi+1 m t i /q i, k t i, i I k, k = 1,, r, y s, y s, A T y + B T z = η, z 0, s 0. Similarly, the constraints s qi i z qi+1 k t i are relaced by the equivalent constraints t ρi i z ρi+1 k s i, where 0 < ρ i := 1/q i 1, and the redundant constraints s 0 are relaced by t 0. The new reformulated dual l -rogramming roblem becomes (DL ) inf c T y + d T z + m t i /q i, s i t ρi i z ρi+1 k, i I k, k = 1,, r, y s, y s, A T y + B T z = η, z 0, s 0. Note that the original roblem (DL ) has r inequalities, and the reformulated roblem (DL ))4m + r. We now have the following results. After doing some straightforward calculations to F (t, z) := t ρ z ρ+1, 0 < ρ < 1, there h T = (h 1, h 2 ), we obtain for the second-order term h T F (t, z)h =ρ(1 ρ)t ρ 3 z ρ 2 (tz 3 h t 3 zh 2 2 2t 2 z 2 h 1 h 2 ) =ρ(1 ρ)t ρ 3 z ρ 2 (zh 1 th 2 ) 2 tz,

11 122 LOU Ye, GAO Yuetian 16ò and for the third-order term 3 F (t, z)[h, h, h] =ρ(1 ρ)t ρ 3 z ρ 2 (ρ 2)z 3 h 3 1 (ρ + 1)t 3 h 3 2 3(ρ 1)tz 2 h 2 1h 2 + 3ρt 2 zh 1 h 3 2 =ρ(1 ρ)t ρ 3 z ρ 2 (zh 1 th 2 ) 2 (ρ 2)zh 1 (ρ + 1)th 2 ρ(1 ρ)(ρ + 1)t ρ 3 z ρ 2 (zh 1 th 2 ) 2 (z h 1 + t h 2 ). Finally we obtain 3 F (t, z)[h, h, h] 2(ρ + 1)h T h1 F (t, z)h 2 t 2 + h 2 2 z 2, where ρ i 1. This exlains log(t ρi i z ρi+1 k s 1 ) log z k log t i is (1, (ρi + 1))-self-concordant. Monteiro and Adler s condition Monteiro and Adler [15] considered minimization roblems with linear equality constraints and a searable convex objective function on the ositive orthant of R n. The objective function f(x) = g i (x i ) must satisfy the following condition. i There exist ositive numbers T and such that for all reals x > 0 and y > 0 and all i = 1,, n, we have y g (3) i (y) T max {( ) x ( } y, y x) g (2) i (x) substituting y = x in the above condition, it is easy to see that g i satisfies γ = T 3, i.e., that the logarithmic barrier function for such a roblem is (1, 1 + T 3 )-self-concordant. Scaled Lischitz condition Interior-oint methods are given and analyzed for roblems with linear equality constraints and convex objective function f(x) on the ositive orthant of R n +. The objective function has to satisfy the following scaled Lischitz condition: There exists M > 0, such that for any ω, 0 < ω < 1, X( f(x + x) f(x) 2 f(x) x) M x T 2 f(x) x (3.2) whenever x > 0 and X 1 x ω. (Here, is the Euclidean norm). This condition is also covered by the self-concordance condition if f is three times continuously differentiable in the interior of the feasible domain. More recisely we can obtain that the corresonding logarithmic barrier function is (1, M)-self-concordant. Since f C 3, we may exand f as follows: f(x + x) = f(x) + 2 f(x) x f(x)[ x, x, ] + o( x 2 ),

12 4Ï Some results of convex rogramming comlexity 123 where f(x)[ x, x, ] is a vector whose ith comonent is equal to j,k 3 f(x) x i x j x k x j x k. Relacing x by λ x in (3.2), inserting the above exansion, dividing by λ 2, and taking the limit λ 0, we obtain X 3 f(x)[ x, x, ] 2M x T 2 f(x) x. Considering X 3 f(x)[ x, x, ] as a column vector, we may continue X 3 f(x)[ x, x, ] (X 1 x) T X 1 x X 3 f(x)[ x, x, ] = 3 f(x)[ x, x, x] X 1 x and obtain that 3 f(x)[ x, x, x] 2M X 1 x x T 2 f(x) x. Let h = x, and notice that = X 1 x, we obtain that the logarithmic barrier function for such a roblem is (1, M)-self-concordant. h 2 i x 2 i 4 Concluding remarks Before we conclude this work, we would like to briefly oint out a class of roblems considered in [16] which does not have a self-concordant logarithmic barrier function. But for most alications however, we believe that the self-concordance condition is more ractical. In this article we resent two different conditions, and we have the conclusion that if the barrier function satisfies the condition, it is also self-concordant. This work rovides us a ath to study convex rogramming roblems, and needed us to study more in the future work. References [1] Nesterov Y E, Nemirovsky A S. Interior oint olynomial algorithms in convex rogramming [J]. SIAM Studies in Alied Mathematics, 1994, 13. [2] Alizadeh E. Otimization over the ositive definite cone: interior-oint methods and combinatorial alications [M]// Advances in Otimization and Parallel Comuting, New York: Elsevier Science Inc, [3] Lustig I J, Mal sten R E, Shanno D E. On imlementing Mehrotra s redictor-corrector interior oint method for linear rogramming [J]. SIAM Journal on Otimization, 1992, 2: [4] Han C G, Pardalos E M, Ye Y. On interior-oint algorithms for some entroy otimization roblems [R]. Technical Reort CS 91-02, Comuter Science Deartment, Pennsylvania State University, Pennsylvania: University Park, PA, 1991.

13 124 LOU Ye, GAO Yuetian 16ò [5] Kortanek K O, No H. A second order affine scaling algorithm for the geometric rogramming dual with logarithmic barrier [J]. Otimization, 1990, 23: [6] Peterson E L, Ecker J G. Geometric rogramming: duality in quadratic rogramming and i aroximation I [C]// Proceedings of the International Symosium of Mathematical Programming. Princeton: Princeton University Press, 1970, [7] Peterson E L, Ecker J G. Geometric rogramming: duality in quadratic rogramming and i aroximation II [J]. SIAM Journal on Alied Mathematics, 1969, 17: [8] Peterson E L, Eeker J G. Geometric rogramming: duality in quadratic rogramming and i aroximation III [J]. Journal of Mathematical Analysis and Alications, 1970, 29: [9] Kortanek K O, Zhu J. A olynomial barrier algorithm for linearly constrained convex rogramming roblems [J]. Mathematics of Oerations Researeh, 1993, 18: [10] Zhu J. A ath following algorithm for a class of convex rogramming roblems [J]. Zeitschrift für Oerations Research, 1992, 36(4): [11] Glineur F. Toics in convex otimization: interior-oint methods, conic deality and aroximations [D]. Belgium: U Mons, [12] Ye Y. Interior algorithms for linear quadratic and linearly constrained convex rogramming [D]. San Francisco: Stanford University, [13] Zhang S. A new self-dual embedding method for convex rogramming [J]. Journal of Global Otimization, 2004, 29: [14] Gao Yuetian, Wu Donghua. A new self-concordant function method for some classes of structured convex rogramming roblems [J]. Alied Mathematical Sciences. [15] Monteiro R D C, Adler I. An extension of Karmarkar tye algorithms to a class of convex searable rogramming roblems with global rate of convergence [J]. Mathematics of Oerations Research, 1989, 15(3): [16] Mehrotra S, Sun J. An interior oint algoritbm for solving smooth convex rograms based on Newton s method [J]. Contemorary Mathematics, 1990, 114:

Notes on duality in second order and -order cone otimization E. D. Andersen Λ, C. Roos y, and T. Terlaky z Aril 6, 000 Abstract Recently, the so-calle

Notes on duality in second order and -order cone otimization E. D. Andersen Λ, C. Roos y, and T. Terlaky z Aril 6, 000 Abstract Recently, the so-calle McMaster University Advanced Otimization Laboratory Title: Notes on duality in second order and -order cone otimization Author: Erling D. Andersen, Cornelis Roos and Tamás Terlaky AdvOl-Reort No. 000/8

More information

Convex Optimization methods for Computing Channel Capacity

Convex Optimization methods for Computing Channel Capacity Convex Otimization methods for Comuting Channel Caacity Abhishek Sinha Laboratory for Information and Decision Systems (LIDS), MIT sinhaa@mit.edu May 15, 2014 We consider a classical comutational roblem

More information

A numerical implementation of a predictor-corrector algorithm for sufcient linear complementarity problem

A numerical implementation of a predictor-corrector algorithm for sufcient linear complementarity problem A numerical imlementation of a redictor-corrector algorithm for sufcient linear comlementarity roblem BENTERKI DJAMEL University Ferhat Abbas of Setif-1 Faculty of science Laboratory of fundamental and

More information

On the Chvatál-Complexity of Knapsack Problems

On the Chvatál-Complexity of Knapsack Problems R u t c o r Research R e o r t On the Chvatál-Comlexity of Knasack Problems Gergely Kovács a Béla Vizvári b RRR 5-08, October 008 RUTCOR Rutgers Center for Oerations Research Rutgers University 640 Bartholomew

More information

arxiv:math/ v4 [math.gn] 25 Nov 2006

arxiv:math/ v4 [math.gn] 25 Nov 2006 arxiv:math/0607751v4 [math.gn] 25 Nov 2006 On the uniqueness of the coincidence index on orientable differentiable manifolds P. Christoher Staecker October 12, 2006 Abstract The fixed oint index of toological

More information

Research Article An iterative Algorithm for Hemicontractive Mappings in Banach Spaces

Research Article An iterative Algorithm for Hemicontractive Mappings in Banach Spaces Abstract and Alied Analysis Volume 2012, Article ID 264103, 11 ages doi:10.1155/2012/264103 Research Article An iterative Algorithm for Hemicontractive Maings in Banach Saces Youli Yu, 1 Zhitao Wu, 2 and

More information

Estimation of the large covariance matrix with two-step monotone missing data

Estimation of the large covariance matrix with two-step monotone missing data Estimation of the large covariance matrix with two-ste monotone missing data Masashi Hyodo, Nobumichi Shutoh 2, Takashi Seo, and Tatjana Pavlenko 3 Deartment of Mathematical Information Science, Tokyo

More information

Multi-Operation Multi-Machine Scheduling

Multi-Operation Multi-Machine Scheduling Multi-Oeration Multi-Machine Scheduling Weizhen Mao he College of William and Mary, Williamsburg VA 3185, USA Abstract. In the multi-oeration scheduling that arises in industrial engineering, each job

More information

Feedback-error control

Feedback-error control Chater 4 Feedback-error control 4.1 Introduction This chater exlains the feedback-error (FBE) control scheme originally described by Kawato [, 87, 8]. FBE is a widely used neural network based controller

More information

Approximating min-max k-clustering

Approximating min-max k-clustering Aroximating min-max k-clustering Asaf Levin July 24, 2007 Abstract We consider the roblems of set artitioning into k clusters with minimum total cost and minimum of the maximum cost of a cluster. The cost

More information

NONLINEAR OPTIMIZATION WITH CONVEX CONSTRAINTS. The Goldstein-Levitin-Polyak algorithm

NONLINEAR OPTIMIZATION WITH CONVEX CONSTRAINTS. The Goldstein-Levitin-Polyak algorithm - (23) NLP - NONLINEAR OPTIMIZATION WITH CONVEX CONSTRAINTS The Goldstein-Levitin-Polya algorithm We consider an algorithm for solving the otimization roblem under convex constraints. Although the convexity

More information

Location of solutions for quasi-linear elliptic equations with general gradient dependence

Location of solutions for quasi-linear elliptic equations with general gradient dependence Electronic Journal of Qualitative Theory of Differential Equations 217, No. 87, 1 1; htts://doi.org/1.14232/ejqtde.217.1.87 www.math.u-szeged.hu/ejqtde/ Location of solutions for quasi-linear ellitic equations

More information

Elementary Analysis in Q p

Elementary Analysis in Q p Elementary Analysis in Q Hannah Hutter, May Szedlák, Phili Wirth November 17, 2011 This reort follows very closely the book of Svetlana Katok 1. 1 Sequences and Series In this section we will see some

More information

Sharp gradient estimate and spectral rigidity for p-laplacian

Sharp gradient estimate and spectral rigidity for p-laplacian Shar gradient estimate and sectral rigidity for -Lalacian Chiung-Jue Anna Sung and Jiaing Wang To aear in ath. Research Letters. Abstract We derive a shar gradient estimate for ositive eigenfunctions of

More information

4. Score normalization technical details We now discuss the technical details of the score normalization method.

4. Score normalization technical details We now discuss the technical details of the score normalization method. SMT SCORING SYSTEM This document describes the scoring system for the Stanford Math Tournament We begin by giving an overview of the changes to scoring and a non-technical descrition of the scoring rules

More information

Asymptotically Optimal Simulation Allocation under Dependent Sampling

Asymptotically Optimal Simulation Allocation under Dependent Sampling Asymtotically Otimal Simulation Allocation under Deendent Samling Xiaoing Xiong The Robert H. Smith School of Business, University of Maryland, College Park, MD 20742-1815, USA, xiaoingx@yahoo.com Sandee

More information

An Infeasible Interior-Point Algorithm with full-newton Step for Linear Optimization

An Infeasible Interior-Point Algorithm with full-newton Step for Linear Optimization An Infeasible Interior-Point Algorithm with full-newton Step for Linear Optimization H. Mansouri M. Zangiabadi Y. Bai C. Roos Department of Mathematical Science, Shahrekord University, P.O. Box 115, Shahrekord,

More information

Multiplicity of weak solutions for a class of nonuniformly elliptic equations of p-laplacian type

Multiplicity of weak solutions for a class of nonuniformly elliptic equations of p-laplacian type Nonlinear Analysis 7 29 536 546 www.elsevier.com/locate/na Multilicity of weak solutions for a class of nonuniformly ellitic equations of -Lalacian tye Hoang Quoc Toan, Quô c-anh Ngô Deartment of Mathematics,

More information

Various Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various Families of R n Norms and Some Open Problems

Various Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various Families of R n Norms and Some Open Problems Int. J. Oen Problems Comt. Math., Vol. 3, No. 2, June 2010 ISSN 1998-6262; Coyright c ICSRS Publication, 2010 www.i-csrs.org Various Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various

More information

3.4 Design Methods for Fractional Delay Allpass Filters

3.4 Design Methods for Fractional Delay Allpass Filters Chater 3. Fractional Delay Filters 15 3.4 Design Methods for Fractional Delay Allass Filters Above we have studied the design of FIR filters for fractional delay aroximation. ow we show how recursive or

More information

arxiv: v2 [math.na] 6 Apr 2016

arxiv: v2 [math.na] 6 Apr 2016 Existence and otimality of strong stability reserving linear multiste methods: a duality-based aroach arxiv:504.03930v [math.na] 6 Ar 06 Adrián Németh January 9, 08 Abstract David I. Ketcheson We rove

More information

Convex Analysis and Economic Theory Winter 2018

Convex Analysis and Economic Theory Winter 2018 Division of the Humanities and Social Sciences Ec 181 KC Border Conve Analysis and Economic Theory Winter 2018 Toic 16: Fenchel conjugates 16.1 Conjugate functions Recall from Proosition 14.1.1 that is

More information

OPTIMAL AFFINE INVARIANT SMOOTH MINIMIZATION ALGORITHMS

OPTIMAL AFFINE INVARIANT SMOOTH MINIMIZATION ALGORITHMS 1 OPTIMAL AFFINE INVARIANT SMOOTH MINIMIZATION ALGORITHMS ALEXANDRE D ASPREMONT, CRISTÓBAL GUZMÁN, AND MARTIN JAGGI ABSTRACT. We formulate an affine invariant imlementation of the accelerated first-order

More information

IMPROVED BOUNDS IN THE SCALED ENFLO TYPE INEQUALITY FOR BANACH SPACES

IMPROVED BOUNDS IN THE SCALED ENFLO TYPE INEQUALITY FOR BANACH SPACES IMPROVED BOUNDS IN THE SCALED ENFLO TYPE INEQUALITY FOR BANACH SPACES OHAD GILADI AND ASSAF NAOR Abstract. It is shown that if (, ) is a Banach sace with Rademacher tye 1 then for every n N there exists

More information

Boundary regularity for elliptic problems with continuous coefficients

Boundary regularity for elliptic problems with continuous coefficients Boundary regularity for ellitic roblems with continuous coefficients Lisa Beck Abstract: We consider weak solutions of second order nonlinear ellitic systems in divergence form or of quasi-convex variational

More information

A Social Welfare Optimal Sequential Allocation Procedure

A Social Welfare Optimal Sequential Allocation Procedure A Social Welfare Otimal Sequential Allocation Procedure Thomas Kalinowsi Universität Rostoc, Germany Nina Narodytsa and Toby Walsh NICTA and UNSW, Australia May 2, 201 Abstract We consider a simle sequential

More information

On Wald-Type Optimal Stopping for Brownian Motion

On Wald-Type Optimal Stopping for Brownian Motion J Al Probab Vol 34, No 1, 1997, (66-73) Prerint Ser No 1, 1994, Math Inst Aarhus On Wald-Tye Otimal Stoing for Brownian Motion S RAVRSN and PSKIR The solution is resented to all otimal stoing roblems of

More information

Supplement: Universal Self-Concordant Barrier Functions

Supplement: Universal Self-Concordant Barrier Functions IE 8534 1 Supplement: Universal Self-Concordant Barrier Functions IE 8534 2 Recall that a self-concordant barrier function for K is a barrier function satisfying 3 F (x)[h, h, h] 2( 2 F (x)[h, h]) 3/2,

More information

HENSEL S LEMMA KEITH CONRAD

HENSEL S LEMMA KEITH CONRAD HENSEL S LEMMA KEITH CONRAD 1. Introduction In the -adic integers, congruences are aroximations: for a and b in Z, a b mod n is the same as a b 1/ n. Turning information modulo one ower of into similar

More information

On a class of Rellich inequalities

On a class of Rellich inequalities On a class of Rellich inequalities G. Barbatis A. Tertikas Dedicated to Professor E.B. Davies on the occasion of his 60th birthday Abstract We rove Rellich and imroved Rellich inequalities that involve

More information

Additive results for the generalized Drazin inverse in a Banach algebra

Additive results for the generalized Drazin inverse in a Banach algebra Additive results for the generalized Drazin inverse in a Banach algebra Dragana S. Cvetković-Ilić Dragan S. Djordjević and Yimin Wei* Abstract In this aer we investigate additive roerties of the generalized

More information

Preconditioning techniques for Newton s method for the incompressible Navier Stokes equations

Preconditioning techniques for Newton s method for the incompressible Navier Stokes equations Preconditioning techniques for Newton s method for the incomressible Navier Stokes equations H. C. ELMAN 1, D. LOGHIN 2 and A. J. WATHEN 3 1 Deartment of Comuter Science, University of Maryland, College

More information

Positive decomposition of transfer functions with multiple poles

Positive decomposition of transfer functions with multiple poles Positive decomosition of transfer functions with multile oles Béla Nagy 1, Máté Matolcsi 2, and Márta Szilvási 1 Deartment of Analysis, Technical University of Budaest (BME), H-1111, Budaest, Egry J. u.

More information

Applied Mathematics and Computation

Applied Mathematics and Computation Alied Mathematics and Comutation 217 (2010) 1887 1895 Contents lists available at ScienceDirect Alied Mathematics and Comutation journal homeage: www.elsevier.com/locate/amc Derivative free two-oint methods

More information

An Estimate For Heilbronn s Exponential Sum

An Estimate For Heilbronn s Exponential Sum An Estimate For Heilbronn s Exonential Sum D.R. Heath-Brown Magdalen College, Oxford For Heini Halberstam, on his retirement Let be a rime, and set e(x) = ex(2πix). Heilbronn s exonential sum is defined

More information

1 Riesz Potential and Enbeddings Theorems

1 Riesz Potential and Enbeddings Theorems Riesz Potential and Enbeddings Theorems Given 0 < < and a function u L loc R, the Riesz otential of u is defined by u y I u x := R x y dy, x R We begin by finding an exonent such that I u L R c u L R for

More information

where x i is the ith coordinate of x R N. 1. Show that the following upper bound holds for the growth function of H:

where x i is the ith coordinate of x R N. 1. Show that the following upper bound holds for the growth function of H: Mehryar Mohri Foundations of Machine Learning Courant Institute of Mathematical Sciences Homework assignment 2 October 25, 2017 Due: November 08, 2017 A. Growth function Growth function of stum functions.

More information

Best approximation by linear combinations of characteristic functions of half-spaces

Best approximation by linear combinations of characteristic functions of half-spaces Best aroximation by linear combinations of characteristic functions of half-saces Paul C. Kainen Deartment of Mathematics Georgetown University Washington, D.C. 20057-1233, USA Věra Kůrková Institute of

More information

Efficient algorithms for the smallest enclosing ball problem

Efficient algorithms for the smallest enclosing ball problem Efficient algorithms for the smallest enclosing ball roblem Guanglu Zhou, Kim-Chuan Toh, Jie Sun November 27, 2002; Revised August 4, 2003 Abstract. Consider the roblem of comuting the smallest enclosing

More information

Robust Solutions to Markov Decision Problems

Robust Solutions to Markov Decision Problems Robust Solutions to Markov Decision Problems Arnab Nilim and Laurent El Ghaoui Deartment of Electrical Engineering and Comuter Sciences University of California, Berkeley, CA 94720 nilim@eecs.berkeley.edu,

More information

Sums of independent random variables

Sums of independent random variables 3 Sums of indeendent random variables This lecture collects a number of estimates for sums of indeendent random variables with values in a Banach sace E. We concentrate on sums of the form N γ nx n, where

More information

Positivity, local smoothing and Harnack inequalities for very fast diffusion equations

Positivity, local smoothing and Harnack inequalities for very fast diffusion equations Positivity, local smoothing and Harnack inequalities for very fast diffusion equations Dedicated to Luis Caffarelli for his ucoming 60 th birthday Matteo Bonforte a, b and Juan Luis Vázquez a, c Abstract

More information

E-companion to A risk- and ambiguity-averse extension of the max-min newsvendor order formula

E-companion to A risk- and ambiguity-averse extension of the max-min newsvendor order formula e-comanion to Han Du and Zuluaga: Etension of Scarf s ma-min order formula ec E-comanion to A risk- and ambiguity-averse etension of the ma-min newsvendor order formula Qiaoming Han School of Mathematics

More information

A full-newton step infeasible interior-point algorithm for linear programming based on a kernel function

A full-newton step infeasible interior-point algorithm for linear programming based on a kernel function A full-newton step infeasible interior-point algorithm for linear programming based on a kernel function Zhongyi Liu, Wenyu Sun Abstract This paper proposes an infeasible interior-point algorithm with

More information

A Note on Guaranteed Sparse Recovery via l 1 -Minimization

A Note on Guaranteed Sparse Recovery via l 1 -Minimization A Note on Guaranteed Sarse Recovery via l -Minimization Simon Foucart, Université Pierre et Marie Curie Abstract It is roved that every s-sarse vector x C N can be recovered from the measurement vector

More information

REFINED STRAIN ENERGY OF THE SHELL

REFINED STRAIN ENERGY OF THE SHELL REFINED STRAIN ENERGY OF THE SHELL Ryszard A. Walentyński Deartment of Building Structures Theory, Silesian University of Technology, Gliwice, PL44-11, Poland ABSTRACT The aer rovides information on evaluation

More information

Statics and dynamics: some elementary concepts

Statics and dynamics: some elementary concepts 1 Statics and dynamics: some elementary concets Dynamics is the study of the movement through time of variables such as heartbeat, temerature, secies oulation, voltage, roduction, emloyment, rices and

More information

Applications to stochastic PDE

Applications to stochastic PDE 15 Alications to stochastic PE In this final lecture we resent some alications of the theory develoed in this course to stochastic artial differential equations. We concentrate on two secific examles:

More information

The inverse Goldbach problem

The inverse Goldbach problem 1 The inverse Goldbach roblem by Christian Elsholtz Submission Setember 7, 2000 (this version includes galley corrections). Aeared in Mathematika 2001. Abstract We imrove the uer and lower bounds of the

More information

A New Self-Dual Embedding Method for Convex Programming

A New Self-Dual Embedding Method for Convex Programming A New Self-Dual Embedding Method for Convex Programming Shuzhong Zhang October 2001; revised October 2002 Abstract In this paper we introduce a conic optimization formulation to solve constrained convex

More information

On Isoperimetric Functions of Probability Measures Having Log-Concave Densities with Respect to the Standard Normal Law

On Isoperimetric Functions of Probability Measures Having Log-Concave Densities with Respect to the Standard Normal Law On Isoerimetric Functions of Probability Measures Having Log-Concave Densities with Resect to the Standard Normal Law Sergey G. Bobkov Abstract Isoerimetric inequalities are discussed for one-dimensional

More information

A PRIORI ESTIMATES AND APPLICATION TO THE SYMMETRY OF SOLUTIONS FOR CRITICAL

A PRIORI ESTIMATES AND APPLICATION TO THE SYMMETRY OF SOLUTIONS FOR CRITICAL A PRIORI ESTIMATES AND APPLICATION TO THE SYMMETRY OF SOLUTIONS FOR CRITICAL LAPLACE EQUATIONS Abstract. We establish ointwise a riori estimates for solutions in D 1, of equations of tye u = f x, u, where

More information

1 Extremum Estimators

1 Extremum Estimators FINC 9311-21 Financial Econometrics Handout Jialin Yu 1 Extremum Estimators Let θ 0 be a vector of k 1 unknown arameters. Extremum estimators: estimators obtained by maximizing or minimizing some objective

More information

Improved Bounds on Bell Numbers and on Moments of Sums of Random Variables

Improved Bounds on Bell Numbers and on Moments of Sums of Random Variables Imroved Bounds on Bell Numbers and on Moments of Sums of Random Variables Daniel Berend Tamir Tassa Abstract We rovide bounds for moments of sums of sequences of indeendent random variables. Concentrating

More information

Topic 7: Using identity types

Topic 7: Using identity types Toic 7: Using identity tyes June 10, 2014 Now we would like to learn how to use identity tyes and how to do some actual mathematics with them. By now we have essentially introduced all inference rules

More information

Difference of Convex Functions Programming for Reinforcement Learning (Supplementary File)

Difference of Convex Functions Programming for Reinforcement Learning (Supplementary File) Difference of Convex Functions Programming for Reinforcement Learning Sulementary File Bilal Piot 1,, Matthieu Geist 1, Olivier Pietquin,3 1 MaLIS research grou SUPELEC - UMI 958 GeorgiaTech-CRS, France

More information

Inference for Empirical Wasserstein Distances on Finite Spaces: Supplementary Material

Inference for Empirical Wasserstein Distances on Finite Spaces: Supplementary Material Inference for Emirical Wasserstein Distances on Finite Saces: Sulementary Material Max Sommerfeld Axel Munk Keywords: otimal transort, Wasserstein distance, central limit theorem, directional Hadamard

More information

Applicable Analysis and Discrete Mathematics available online at HENSEL CODES OF SQUARE ROOTS OF P-ADIC NUMBERS

Applicable Analysis and Discrete Mathematics available online at   HENSEL CODES OF SQUARE ROOTS OF P-ADIC NUMBERS Alicable Analysis and Discrete Mathematics available online at htt://efmath.etf.rs Al. Anal. Discrete Math. 4 (010), 3 44. doi:10.98/aadm1000009m HENSEL CODES OF SQUARE ROOTS OF P-ADIC NUMBERS Zerzaihi

More information

LECTURE 7 NOTES. x n. d x if. E [g(x n )] E [g(x)]

LECTURE 7 NOTES. x n. d x if. E [g(x n )] E [g(x)] LECTURE 7 NOTES 1. Convergence of random variables. Before delving into the large samle roerties of the MLE, we review some concets from large samle theory. 1. Convergence in robability: x n x if, for

More information

COMMUNICATION BETWEEN SHAREHOLDERS 1

COMMUNICATION BETWEEN SHAREHOLDERS 1 COMMUNICATION BTWN SHARHOLDRS 1 A B. O A : A D Lemma B.1. U to µ Z r 2 σ2 Z + σ2 X 2r ω 2 an additive constant that does not deend on a or θ, the agents ayoffs can be written as: 2r rθa ω2 + θ µ Y rcov

More information

The Algebraic Structure of the p-order Cone

The Algebraic Structure of the p-order Cone The Algebraic Structure of the -Order Cone Baha Alzalg Abstract We study and analyze the algebraic structure of the -order cones We show that, unlike oularly erceived, the -order cone is self-dual for

More information

A note on variational representation for singular values of matrix

A note on variational representation for singular values of matrix Alied Mathematics and Comutation 43 (2003) 559 563 www.elsevier.com/locate/amc A note on variational reresentation for singular values of matrix Zhi-Hao Cao *, Li-Hong Feng Deartment of Mathematics and

More information

LEIBNIZ SEMINORMS IN PROBABILITY SPACES

LEIBNIZ SEMINORMS IN PROBABILITY SPACES LEIBNIZ SEMINORMS IN PROBABILITY SPACES ÁDÁM BESENYEI AND ZOLTÁN LÉKA Abstract. In this aer we study the (strong) Leibniz roerty of centered moments of bounded random variables. We shall answer a question

More information

Linear diophantine equations for discrete tomography

Linear diophantine equations for discrete tomography Journal of X-Ray Science and Technology 10 001 59 66 59 IOS Press Linear diohantine euations for discrete tomograhy Yangbo Ye a,gewang b and Jiehua Zhu a a Deartment of Mathematics, The University of Iowa,

More information

Finding Shortest Hamiltonian Path is in P. Abstract

Finding Shortest Hamiltonian Path is in P. Abstract Finding Shortest Hamiltonian Path is in P Dhananay P. Mehendale Sir Parashurambhau College, Tilak Road, Pune, India bstract The roblem of finding shortest Hamiltonian ath in a eighted comlete grah belongs

More information

Uniform Law on the Unit Sphere of a Banach Space

Uniform Law on the Unit Sphere of a Banach Space Uniform Law on the Unit Shere of a Banach Sace by Bernard Beauzamy Société de Calcul Mathématique SA Faubourg Saint Honoré 75008 Paris France Setember 008 Abstract We investigate the construction of a

More information

Probability Estimates for Multi-class Classification by Pairwise Coupling

Probability Estimates for Multi-class Classification by Pairwise Coupling Probability Estimates for Multi-class Classification by Pairwise Couling Ting-Fan Wu Chih-Jen Lin Deartment of Comuter Science National Taiwan University Taiei 06, Taiwan Ruby C. Weng Deartment of Statistics

More information

DIFFERENTIAL GEOMETRY. LECTURES 9-10,

DIFFERENTIAL GEOMETRY. LECTURES 9-10, DIFFERENTIAL GEOMETRY. LECTURES 9-10, 23-26.06.08 Let us rovide some more details to the definintion of the de Rham differential. Let V, W be two vector bundles and assume we want to define an oerator

More information

k- price auctions and Combination-auctions

k- price auctions and Combination-auctions k- rice auctions and Combination-auctions Martin Mihelich Yan Shu Walnut Algorithms March 6, 219 arxiv:181.3494v3 [q-fin.mf] 5 Mar 219 Abstract We rovide for the first time an exact analytical solution

More information

Combining Logistic Regression with Kriging for Mapping the Risk of Occurrence of Unexploded Ordnance (UXO)

Combining Logistic Regression with Kriging for Mapping the Risk of Occurrence of Unexploded Ordnance (UXO) Combining Logistic Regression with Kriging for Maing the Risk of Occurrence of Unexloded Ordnance (UXO) H. Saito (), P. Goovaerts (), S. A. McKenna (2) Environmental and Water Resources Engineering, Deartment

More information

On the minimax inequality and its application to existence of three solutions for elliptic equations with Dirichlet boundary condition

On the minimax inequality and its application to existence of three solutions for elliptic equations with Dirichlet boundary condition ISSN 1 746-7233 England UK World Journal of Modelling and Simulation Vol. 3 (2007) No. 2. 83-89 On the minimax inequality and its alication to existence of three solutions for ellitic equations with Dirichlet

More information

GOOD MODELS FOR CUBIC SURFACES. 1. Introduction

GOOD MODELS FOR CUBIC SURFACES. 1. Introduction GOOD MODELS FOR CUBIC SURFACES ANDREAS-STEPHAN ELSENHANS Abstract. This article describes an algorithm for finding a model of a hyersurface with small coefficients. It is shown that the aroach works in

More information

Quantitative estimates of propagation of chaos for stochastic systems with W 1, kernels

Quantitative estimates of propagation of chaos for stochastic systems with W 1, kernels oname manuscrit o. will be inserted by the editor) Quantitative estimates of roagation of chaos for stochastic systems with W, kernels Pierre-Emmanuel Jabin Zhenfu Wang Received: date / Acceted: date Abstract

More information

MODELING THE RELIABILITY OF C4ISR SYSTEMS HARDWARE/SOFTWARE COMPONENTS USING AN IMPROVED MARKOV MODEL

MODELING THE RELIABILITY OF C4ISR SYSTEMS HARDWARE/SOFTWARE COMPONENTS USING AN IMPROVED MARKOV MODEL Technical Sciences and Alied Mathematics MODELING THE RELIABILITY OF CISR SYSTEMS HARDWARE/SOFTWARE COMPONENTS USING AN IMPROVED MARKOV MODEL Cezar VASILESCU Regional Deartment of Defense Resources Management

More information

Rotations in Curved Trajectories for Unconstrained Minimization

Rotations in Curved Trajectories for Unconstrained Minimization Rotations in Curved rajectories for Unconstrained Minimization Alberto J Jimenez Mathematics Deartment, California Polytechnic University, San Luis Obiso, CA, USA 9407 Abstract Curved rajectories Algorithm

More information

THE 2D CASE OF THE BOURGAIN-DEMETER-GUTH ARGUMENT

THE 2D CASE OF THE BOURGAIN-DEMETER-GUTH ARGUMENT THE 2D CASE OF THE BOURGAIN-DEMETER-GUTH ARGUMENT ZANE LI Let e(z) := e 2πiz and for g : [0, ] C and J [0, ], define the extension oerator E J g(x) := g(t)e(tx + t 2 x 2 ) dt. J For a ositive weight ν

More information

arxiv: v1 [math.nt] 11 Jun 2016

arxiv: v1 [math.nt] 11 Jun 2016 ALMOST-PRIME POLYNOMIALS WITH PRIME ARGUMENTS P-H KAO arxiv:003505v [mathnt Jun 20 Abstract We imrove Irving s method of the double-sieve [8 by using the DHR sieve By extending the uer and lower bound

More information

Monopolist s mark-up and the elasticity of substitution

Monopolist s mark-up and the elasticity of substitution Croatian Oerational Research Review 377 CRORR 8(7), 377 39 Monoolist s mark-u and the elasticity of substitution Ilko Vrankić, Mira Kran, and Tomislav Herceg Deartment of Economic Theory, Faculty of Economics

More information

p-adic Measures and Bernoulli Numbers

p-adic Measures and Bernoulli Numbers -Adic Measures and Bernoulli Numbers Adam Bowers Introduction The constants B k in the Taylor series exansion t e t = t k B k k! k=0 are known as the Bernoulli numbers. The first few are,, 6, 0, 30, 0,

More information

Positive Definite Uncertain Homogeneous Matrix Polynomials: Analysis and Application

Positive Definite Uncertain Homogeneous Matrix Polynomials: Analysis and Application BULGARIA ACADEMY OF SCIECES CYBEREICS AD IFORMAIO ECHOLOGIES Volume 9 o 3 Sofia 009 Positive Definite Uncertain Homogeneous Matrix Polynomials: Analysis and Alication Svetoslav Savov Institute of Information

More information

Commutators on l. D. Dosev and W. B. Johnson

Commutators on l. D. Dosev and W. B. Johnson Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Commutators on l D. Dosev and W. B. Johnson Abstract The oerators on l which are commutators are those not of the form λi

More information

On the smallest point on a diagonal quartic threefold

On the smallest point on a diagonal quartic threefold On the smallest oint on a diagonal quartic threefold Andreas-Stehan Elsenhans and Jörg Jahnel Abstract For the family x = a y +a 2 z +a 3 v + w,,, > 0, of diagonal quartic threefolds, we study the behaviour

More information

ON UNIFORM BOUNDEDNESS OF DYADIC AVERAGING OPERATORS IN SPACES OF HARDY-SOBOLEV TYPE. 1. Introduction

ON UNIFORM BOUNDEDNESS OF DYADIC AVERAGING OPERATORS IN SPACES OF HARDY-SOBOLEV TYPE. 1. Introduction ON UNIFORM BOUNDEDNESS OF DYADIC AVERAGING OPERATORS IN SPACES OF HARDY-SOBOLEV TYPE GUSTAVO GARRIGÓS ANDREAS SEEGER TINO ULLRICH Abstract We give an alternative roof and a wavelet analog of recent results

More information

Intrinsic Approximation on Cantor-like Sets, a Problem of Mahler

Intrinsic Approximation on Cantor-like Sets, a Problem of Mahler Intrinsic Aroximation on Cantor-like Sets, a Problem of Mahler Ryan Broderick, Lior Fishman, Asaf Reich and Barak Weiss July 200 Abstract In 984, Kurt Mahler osed the following fundamental question: How

More information

Inequalities for finite trigonometric sums. An interplay: with some series related to harmonic numbers

Inequalities for finite trigonometric sums. An interplay: with some series related to harmonic numbers Kouba Journal of Inequalities and Alications 6 6:73 DOI.86/s366-6-- R E S E A R C H Oen Access Inequalities for finite trigonometric sums. An interlay: with some series related to harmonic numbers Omran

More information

Solvability and Number of Roots of Bi-Quadratic Equations over p adic Fields

Solvability and Number of Roots of Bi-Quadratic Equations over p adic Fields Malaysian Journal of Mathematical Sciences 10(S February: 15-35 (016 Secial Issue: The 3 rd International Conference on Mathematical Alications in Engineering 014 (ICMAE 14 MALAYSIAN JOURNAL OF MATHEMATICAL

More information

Combinatorics of topmost discs of multi-peg Tower of Hanoi problem

Combinatorics of topmost discs of multi-peg Tower of Hanoi problem Combinatorics of tomost discs of multi-eg Tower of Hanoi roblem Sandi Klavžar Deartment of Mathematics, PEF, Unversity of Maribor Koroška cesta 160, 000 Maribor, Slovenia Uroš Milutinović Deartment of

More information

Math 701: Secant Method

Math 701: Secant Method Math 701: Secant Method The secant method aroximates solutions to f(x = 0 using an iterative scheme similar to Newton s method in which the derivative has been relace by This results in the two-term recurrence

More information

Robust hamiltonicity of random directed graphs

Robust hamiltonicity of random directed graphs Robust hamiltonicity of random directed grahs Asaf Ferber Rajko Nenadov Andreas Noever asaf.ferber@inf.ethz.ch rnenadov@inf.ethz.ch anoever@inf.ethz.ch Ueli Peter ueter@inf.ethz.ch Nemanja Škorić nskoric@inf.ethz.ch

More information

Estimation of Separable Representations in Psychophysical Experiments

Estimation of Separable Representations in Psychophysical Experiments Estimation of Searable Reresentations in Psychohysical Exeriments Michele Bernasconi (mbernasconi@eco.uninsubria.it) Christine Choirat (cchoirat@eco.uninsubria.it) Raffaello Seri (rseri@eco.uninsubria.it)

More information

Stochastic integration II: the Itô integral

Stochastic integration II: the Itô integral 13 Stochastic integration II: the Itô integral We have seen in Lecture 6 how to integrate functions Φ : (, ) L (H, E) with resect to an H-cylindrical Brownian motion W H. In this lecture we address the

More information

ON THE NORM OF AN IDEMPOTENT SCHUR MULTIPLIER ON THE SCHATTEN CLASS

ON THE NORM OF AN IDEMPOTENT SCHUR MULTIPLIER ON THE SCHATTEN CLASS PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 000-9939XX)0000-0 ON THE NORM OF AN IDEMPOTENT SCHUR MULTIPLIER ON THE SCHATTEN CLASS WILLIAM D. BANKS AND ASMA HARCHARRAS

More information

On Character Sums of Binary Quadratic Forms 1 2. Mei-Chu Chang 3. Abstract. We establish character sum bounds of the form.

On Character Sums of Binary Quadratic Forms 1 2. Mei-Chu Chang 3. Abstract. We establish character sum bounds of the form. On Character Sums of Binary Quadratic Forms 2 Mei-Chu Chang 3 Abstract. We establish character sum bounds of the form χ(x 2 + ky 2 ) < τ H 2, a x a+h b y b+h where χ is a nontrivial character (mod ), 4

More information

MATH 2710: NOTES FOR ANALYSIS

MATH 2710: NOTES FOR ANALYSIS MATH 270: NOTES FOR ANALYSIS The main ideas we will learn from analysis center around the idea of a limit. Limits occurs in several settings. We will start with finite limits of sequences, then cover infinite

More information

A Numerical Radius Version of the Arithmetic-Geometric Mean of Operators

A Numerical Radius Version of the Arithmetic-Geometric Mean of Operators Filomat 30:8 (2016), 2139 2145 DOI 102298/FIL1608139S Published by Faculty of Sciences and Mathematics, University of Niš, Serbia vailable at: htt://wwwmfniacrs/filomat Numerical Radius Version of the

More information

Some Unitary Space Time Codes From Sphere Packing Theory With Optimal Diversity Product of Code Size

Some Unitary Space Time Codes From Sphere Packing Theory With Optimal Diversity Product of Code Size IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 5, NO., DECEMBER 4 336 Some Unitary Sace Time Codes From Shere Packing Theory With Otimal Diversity Product of Code Size Haiquan Wang, Genyuan Wang, and Xiang-Gen

More information

HEAT AND LAPLACE TYPE EQUATIONS WITH COMPLEX SPATIAL VARIABLES IN WEIGHTED BERGMAN SPACES

HEAT AND LAPLACE TYPE EQUATIONS WITH COMPLEX SPATIAL VARIABLES IN WEIGHTED BERGMAN SPACES Electronic Journal of ifferential Equations, Vol. 207 (207), No. 236,. 8. ISSN: 072-669. URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu HEAT AN LAPLACE TYPE EQUATIONS WITH COMPLEX SPATIAL

More information

Nonsymmetric potential-reduction methods for general cones

Nonsymmetric potential-reduction methods for general cones CORE DISCUSSION PAPER 2006/34 Nonsymmetric potential-reduction methods for general cones Yu. Nesterov March 28, 2006 Abstract In this paper we propose two new nonsymmetric primal-dual potential-reduction

More information

Approximation of the Euclidean Distance by Chamfer Distances

Approximation of the Euclidean Distance by Chamfer Distances Acta Cybernetica 0 (0 399 47. Aroximation of the Euclidean Distance by Chamfer Distances András Hajdu, Lajos Hajdu, and Robert Tijdeman Abstract Chamfer distances lay an imortant role in the theory of

More information

Conversions among Several Classes of Predicate Encryption and Applications to ABE with Various Compactness Tradeoffs

Conversions among Several Classes of Predicate Encryption and Applications to ABE with Various Compactness Tradeoffs Conversions among Several Classes of Predicate Encrytion and Alications to ABE with Various Comactness Tradeoffs Nuttaong Attraadung, Goichiro Hanaoka, and Shota Yamada National Institute of Advanced Industrial

More information