An Interior-Point Trust-Region Algorithm for Quadratic Stochastic Symmetric Programming

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1 Thai Journal of Mathematics Volume 5 07 Number : ISSN An Interior-Point Trust-Region Algorithm for Quadratic Stochastic Symmetric Programming Phannipa Kabcome and Thanasak Mouktonglang Department of Mathematics, Faculty of Science Chiang Mai University Chiang Mai 5000, Thailand phannipa.math@gmail.com P. Kabcome thanasak@chiangmai.ac.th T. Mouktonglang Abstract : Stochastic programming is a framework for modeling optimization problems that involve uncertainty. In this paper, we study two-stage stochastic quadratic symmetric programming to handle uncertainty in data defining Deterministic symmetric programs in which a quadratic function is minimized over the intersection of an affine set and a symmetric cone with finite event space. Twostage stochastic programs can be modeled as large deterministic programming and we present an interior point trust region algorithm to solve this problem. Numerical results on randomly generated data are available for stochastic symmetric programs. The complexity of our algorithm is proved. Keywords : interior-point algorithm; trust-region subproblem; symmetric programming; Stochastic programming. 00 Mathematics Subject Classification : 90C30; 90C5. Introduction Deterministic optimization problems are formulated to find optimal solutions in problems with certainty in data. In fact, in some applications we cannot specify the model entirely because it depends on information which is not available at the time of formulation but that will be determined at some point in the future. Corresponding author. Copyright c 07 by the Mathematical Association of Thailand. All rights reserved.

2 38 Thai J. Math. 5 07/ P. Kabcome and T. Mouktonglang Stochastic programming began in the 950s to handle those problems that involve uncertainty in data. Many real world applications carry an inherent uncertainty within them. In particular, two-stage stochastic programs have been established to formulate many applications to linear, non-linear and integer programming. Conic programming deals with an important class of tractable convex optimization problems. Symmetric programming is a generalization of linear programming that includes second-order cone programming and semidefinite programming as special cases. All of these problems are treated in a unified theoretical framework for the design of algorithms and the analysis of their complexity. Stochastic symmetric programming may be viewed as an extension of symmetric programming by allowing uncertainty in data. Interior point methods are considered to be one of the most successful classes of algorithms for solving convex optimization problems. We refer to [] which developed an interior-point trust-region algorithm for minimizing a quadratic objective function in the intersection of a symmetric cone and an affine subspace by solving sequential trust-region subproblems. Global first-order and second-order convergence results were proved. The techniques and properties in both interior-point algorithms and trust-region methods show the complexity of the algorithm by providing strong theoretical support. Our algorithm is developed as an interior-point trust-region algorithm for minimizing stochastic symmetric programming. In this paper, we present stochastic symmetric programming with a quadratic function and finite scenarios. And then we will explicitly formulate a problem as a large scale deterministic program and develop an interior-point trust-region algorithm to solve it. The organization of this paper is as follows. In section, we present some concepts and results of symmetric cone and self-concordant barrier function used in the theory of interior-point method. In section 3, we propose a model of twostage stochastic symmetric programs with linear and quadratic function. We also describe a model in deterministic programming. In section 4, we present an interior point trust-region algorithm for solving stochastic symmetric programming and provide complexity of our algorithm. In section 5, we present computational results on a case study problem. In addition, we compared our results with those of the quadprog function in the MATLAB optimization toolbox in cases of the nonnegative orthant cones. Our algorithm can solve stochastic symmetric programming problems that include variables in second order cone and cones of real symmetric positive semidefinite matrices. The quadprog function is unable to solve problems with these variables. Section 6 has some conclusion and remarks. Symmetric Cone and Self-Concordant Barrier In this section, we introduce some of the concepts and relevant results that will be useful in section 4. For more detailed exposure to those concepts, see the reference [] and [3]. Let E be finite-dimensional real Euclidean space with inner product,.

3 An Interior-Point Trust-Region Algorithm for Quadratic Definition. Cone. A subset K of E is said to be a cone if for every x K and λ > 0 imply that λx K. Definition. Convex cone. A subset K of E is a convex cone if and only if for every x,y K and λ,µ > 0 imply that λxµy K. We assume that K is a convex cone in a E. The dual of K is defined as Definition.3 Dual cone. Let K be a cone. The set is called the dual cone of K. K = {y E x,y 0, x K}. As the name suggests, K is a cone, and is always convex, even when the original cone is not. If cone K and its dual K coincide, we say that K is self-dual. In particular, this implies that K has a nonempty interior and does not contain any straight line i.e., it is pointed. We denote by GLE, the group of general linear transformations on E, and by AutK, the automorphism group of convex cone K, that is AutK = {g GLE gk = K}. Definition.4 Homogeneity. The convex cone K is said to be homogeneous if for every pair x,y intk, there exists an invertible linear operator g for which gk = K and gx = y. Definition.5 Symmetric cone. The convex cone K is said to be symmetric if it is self-dual and homogeneous. Almost all conic optimization problems in real world applications are associated with symmetric cones such as nonnegative orthant cones, second-order cones and cone of positive semi-definite matrices over the real or complex numbers. The nonnegative orthant cone R n = {x R n x i 0,i =,,...,n}. The second-order cone also known as the quadratic, Lorentz, or the ice cream cone of dimension n n Q n = {x R n x i x n and x n 0}. i= The positive semi-definite cone S n = {X X Rn n,x is positive semidefinite matric}. A self-concordant barrier function is crucial to an interior-point methods. We introduce some of the fundamentals idea a self-concordant barrier function that will be useful in this paper.

4 40 Thai J. Math. 5 07/ P. Kabcome and T. Mouktonglang Definition.6 Self-Concordant Function. [3] Let K 0 K E and F :K 0 R be a C 3 smooth convex function such that F is called self-concordant function if F x[h,h,h] F xh,h 3/. for all x K 0 and for all h S. Definition.7 Self-concordant barrier. [3] A self-concordant function F is a self-concordant barrier for a closed convex set K if ϑ = sup x K 0 F x,f x F x. the value ϑ is called a barrier parameter of F. In principle, every convex cone admits a self-concordant barriersee[3], section 4. We consider a self-concordant barrier of symmetric cones as follows Fx = n lnx i i= Fx = lnx n n Fx = lndetx x i i= with ϑ = n, for cone of nonnegative orthant, with ϑ =, for second-order cone, with ϑ = n, for positive semidefinite cone. Lemma.. [4] Let Fx is a self-concordant barrier for K, then F x F x = x,.3 F x,x = ϑ..4 Lemma.. [4] If K Fx is a symmetric cone and F is a self-concordant barrier for K, then F x is a linear automorphism of K for each x K 0. The strictly convex assumption of Fx implies that F x is positive definite for every x K 0. This allows us to define a norm as follows h x = h,f xh.5 is a norm on E induced by F x. Let B x y,r denote the open ball of radius r centered at y, where the radius is measured with respect to. x. Lemma.3. [4] If Fx is a self-concordant function for K, then we have B x x, K 0 for all x K 0. Lemma.4. [4] Assume Fx is a self-concordant function for K, x K 0, and y B x x,, then Fy Fx F x,y x y x,f xy x y x 3 x 3 y x x..6

5 An Interior-Point Trust-Region Algorithm for Quadratic... 4 Lemma.5. [4] Assume Fx is a self-concordant function for K. If nx x /4, then Fx has a minimizer z and z x x nx x where nx = F x F x be the Newton step of Fx. 3 nx x nx x 3,.7 Lemma.6. [4] Assume Fx is a self-concordant barrier with barrier parameter ϑ. If x,y K 0, then F x,y x ϑ..8 3 The Problem and Its Modeling In a standard two-stage stochastic programming model, decision variables are divided into two subsets. First-stage variables are groups of variables determined before the realizations of random events are known. Once the uncertain events have unfolded, further design or operational adjustments can be made through values of the second-stage known as recourse variables, which are determined after knowing the realized values of the random events. A standard formulation of the two-stage stochastic program is min f xe[qx,ξ] 3. st. A 0 x = b, 3. x K, 3.3 where x is the first-stage decision variable, Qx, ξ is the optimal value of the second-stage problem: min f y,ξ 3.4 st. TξxWξy = hξ, 3.5 y K 3.6 where y is the second-stagevariable and the cones K and K are symmetric cones. The matrix A 0 and the vectors b and c are deterministic data, the matrices Wξ and Tξ and the vectorshξ and dξ arerandom data whoserealizationsdepend on an underlying outcome ξ with a known probability function p. 3. The Stochastic Linear Symmetric Programs SLSP The stochastic linear symmetric programs can be stated as: min c, x E[Qx, ξ] 3.7 st. A 0 x = b, 3.8 x K, 3.9

6 4 Thai J. Math. 5 07/ P. Kabcome and T. Mouktonglang where x is the first-stage decision variable, Qx,ξ is the minimum of the problem min gξ, y 3.0 st. TξxWξy = hξ, 3. y K 3. where y is the second-stage variable and the cones K and K are symmetric cones. The formulation of the above problem assumes that the second-stage data ξ can be modeled as a random vector with a known probability distribution. The random vector ξ has a finite number of possible realizations, called scenarios, say ξ,...,ξ K with respective probability masses p,...,p K. Then the expectation in the first-stage problem s objective function can be written as the summation E[Qx,ξ] = p k Qx,ξ k. 3.3 A SLSP can be written as a deterministic program and represented as follows: min qx,y,y,...,y K = c,x K p k g k,y k 3.4 st. A 0 x = b, 3.5 T k xw k y k = h k,k =,,...,K, 3.6 x K,y k K,k =,,...,K, 3.7 where x R n is the first-stage decision variable and y k R k are the second stage variable. The matrix A 0 R m n, the vectors b R m and c R n are deterministic data. The matrices W k R m k n k, T k R m k n and vectors h k R n k, g k R n k are random data associated with probability for k =,,...,K. The cones K and K are symmetric cone in R n and R n k for k =,,...,K n,n k are positive integers. 3. The Stochastic Quadratic Symmetric Programs SQSP The stochastic quadratic symmetric programs can also be cast as SLSP. The matrix H 0 R n n,h 0 0 is deterministic data and the matrices H ξ is random data whose realizations depend on an underlying outcome ξ with a known probability function p. Let H k R n k n k,h k 0 be random data associated with probability for k =,,...,K. A SQSP can be formulated as min x,h 0xc,xE[Qx,ξ] 3.8 st.a 0 x = b, 3.9 x K, 3.0

7 An Interior-Point Trust-Region Algorithm for Quadratic where x is the first-stage decision variable, and Qx,ξ is the minimum of the problem min y,h ξygξ,y 3. st.tξxwξy = hξ, 3. y K, 3.3 where y is the second-stagevariable and the cones K and K are symmetric cones. A SQSP can be equivalently written as a deterministic symmetric programs as: Define min q x,y,y,...,y K = x,h 0xc,x K p k yk,h k yk g k,y k 3.4 st. A 0 x = b, 3.5 T k xw k y k = h k,k =,,...,K, 3.6 x K,y k K,k =,,...,K. 3.7 K = K K K...,K K is the symmetric cone, 3.8 X=[x,y,y,,y K ] T R n k n,x K,y k K,for,,...,K, 3.9 B = [ b, h, h,, h K] T m k R m, 3.30 C = [ c,p g,p g,,p K g K] T n k R n, 3.3 A T W 0 0 A = T 0 W 0 R m m k n K k n, 3.3. : :.. T K 0 0 W K H p H 0 0 Q = 0 0 p H 0 R n n k n K k n,. : : p K H K 3.33

8 44 Thai J. Math. 5 07/ P. Kabcome and T. Mouktonglang we can rewrite a SQSP as a large deterministic program as follow minqx = X,QXC,X 3.34 st.ax = B, 3.35 X K The Interior-Point Trust-Region Algorithm In this section, we present our interior-point trust-region algorithm for solving First, we define the function as f ηi X = η i qxfx = η i x,h 0xc,x F x p k yk,h k y k g k,y k F y k 4. where FX : K R is a self-concordant barrier for the symmetric cone K with a barrier parameter ϑ. A self-concordant barrier define by FX = F x F y k where F x is a self-concordant barrier function for cone K with a barrier parameter ϑ and F y k is a self-concordant barrier function for cone K k with a barrier parameter ϑ k for k =,,3,...,K and ϑ = ϑ K ϑ k SinceqXisquadraticandfromdefinition.6thenf ηi isalsoaself-concordant barrier function. Consider interior-point trust-region algorithm in each inner iteration for a fix η i we need to find the central path for decreased the value of f ηi X. For outer iterations we need increase the value of η i to positive infinity, which implies that the central path converges to a solution that is optimal for the original problem. Let d = d,d,d,d3,,dk E be the trail step of fηi. Consider function f ηi in each inner iteration f ηi x i,j d,y i,j d,y i,j d,...,yk i,j d K f ηi x i,j,y i,j,y i,j,...,yk = η i qx i,j d,y i,j d,y i,j d,...,yk i,j d K F x i,j d F y k i,j dk η i qx i,j,y i,j,y i,j,...,yk i,j F x i,j F y k i,j. i,j

9 An Interior-Point Trust-Region Algorithm for Quadratic = η i xi,j d,h 0 x i,j d η i c,x i,j d F x i,j d η k ip k y i,j dk,hk yk i,j dk η i p k g k,y k i,j dk Fk y k i,j dk η i xi,j,h 0 x i,j ηi c,x i,j F x i,j η i p k g k,y k i,j Fk y k i,j = d,η i H 0 d η i H 0 x i,j c,d η k ip k y dk,η ip k H k ηi p k H k y k i,j gk,d k F x i,j d F x i,j i,j,hk y k i,j dk F y k i,j dk F y k i,j. 4. Wewanttofind theboundoff ηi x i,j d,y i,j d,y i,j d,...,yk i,j dk then we consider the self-concordant barrier in above equation. From Lemma.3, for any X i,j K 0, we have X i,j d K 0. Given F x i,j d α i,j <, F y k i,j d k αk < for k =,,...,K and α i,j K α k <. Implied that F x i,j d K Let F X = F x K F yk i,j d k α i,j K α k <. F y k denote the Hessian of a self-concordant function FX. Since it is positive definite for every X K 0 implies that F X is positive definite. For h = [h,h,h,,hk ] E. Let further we define a norm on E induced by F X as h X = h,f Xh = h,f xh h k,f yk h k F x i,j d F x i,j F y k i,j dk F y k i,j F x i,j,d d,f x d 3 x i,jd i,j 3 d xi,j k K d,f yk i,j dk d k 3 y k i,j 3 d k y k i,j F y k i,j,dk

10 46 Thai J. Math. 5 07/ P. Kabcome and T. Mouktonglang F x i,j,d d,f K x i,j d F y k k d,f y k i,j dk α 3 i,j 3 α i,j F x i,j,d d,f K x i,j d F y k k d,f y k i,j dk i,j,dk α k 3 α k i,j,dk α i,j K α k 3 3 α i,j K α k. 4.3 The third inequality of 4.3 follows from Lemma.4. From 4.3 we can rewrite 4. as f ηi x i,j d,y i,j d,y i,j d,...,yk i,j d K f ηi x i,j,y i,j,y i,j,...,yk d,η i H 0 d η i H 0 x i,j c,d,η ip k H k d k F x i,j,d d,f x K i,jd = 3 k d,f yk i,j dk dk F yk i,j,dk α i,j K α k 3 3 α i,j K α k d, η i H 0 F x i,j d η i H 0 x i,j cf x i,j,d d k, η i p k H k F y k i,j d k α i,j K α i,j K α k 3 α k i,j k η i p k H yk i,j gk F y k i,j,dk. 4.4 d Nowtheinequality4.4givesanupperboundforf ηi x i,j d,y i,j d,y i,j,...,yk. We can try to minimize this bound given in the right-hand side of inequality. This lead to the following trust-region subproblem i,j d K

11 An Interior-Point Trust-Region Algorithm for Quadratic min d, η i H 0 F x i,j d η i H 0 x i,j cf x i,j,d d k, η i p k H k F y k i,j d k k η i p k H y k i,j gk F yk i,j,dk = m i,j d,d,d,...,dk 4.5 st. F x i,j d Let the transformation F y k i,j d k α i,j α k. 4.6 d = F x i,j d, 4.7 and define d k = F yk k i,j d,for k =,,...,K, 4.8 Q i,j = η i F x i,j H0 F x i,j I, 4.9 C i,j = F x i,j ηi H 0 x i,j cf x i,j, 4.0 Q k = η i p k F y k i,j k H F y k i,j I k,for k =,,...,K, 4. C k = F yk i,j ηi p k H k y k i,j gk F yk i,j,for k =,,...,K. 4. We can rewrite equations 4.5 and 4.6 as follow minq i,j d,d,d,...,d K = d,q i,j d C i,j,d K d k k,q d k k C,d k 4.3 st. d d k α i,j α k. 4.4 Once d i,j is computed then we obtain the trail step d i,j = F x i,j d i,j, 4.5

12 48 Thai J. Math. 5 07/ P. Kabcome and T. Mouktonglang d k = F yk i,j d k for k =,,...,K 4.6 and from inequality 4.4 that we have f ηi x i,j d i,j,y i,j d,...,y K i,j d K f ηi x i,j,y i,j,y i,j,...,yk i,j q i,j d i,j,d,...,d K α i,j K α k 3 i,j 3 α i,j K α k. 4.7 Let n ηi X i,j = n ηi x i,j,y i,j,...,yk i,j be the Newton step of f η i x,y,...,y K at the point x i,j,y i,j,y i,j,...,yk i,j. We should point out that n ηi X i,j X i,j = f η i X i,j f η i X i,j,f η i X i,j f η i X i,j f η i X i,j = f η i X i,j,f η i X i,j f η i X i,j = η i H 0 x i,j cf x i,j,η i H 0 F x i,j η i H 0 x i,j cf x i,j η i p k H k yk i,j gk F yk i,j, η i p k H k F yk i,j η i p k H k y k i,j gk F yk i,j = C i,j,q i,j C i,j C k,q k C k 4.8 where the last equality follows equalities Now we are ready to present our algorithm. From what we defined in 3.9, 3.30 and 3.3, we define the feasible set by F = {X E AX = B,x K,y k K, for k =,,...,K} Algorithm : An interior-point trust-region algorithm Step 0 Initialization seti = 0,j = 0, choose starting point x 0,0,y 0,0,y 0,0,...,yK 0,0 {F}, an initial trust-region radius α 0,0 0,, and an initial parameter η 0 are given. Until convergence repeat. Step Test inner iteration termination compute n ηi X i,j X i,j

13 An Interior-Point Trust-Region Algorithm for Quadratic if n ηi X i,j X i,j go to step, otherwise set 36 x i,j,y i,j,y i,j,...,yk i,j = x i,j,y i,j,y i,j,...,yk and go to step 3. i,j Step Step calculation solve 4.3and 4.4 to obtain d i,j,d,d,...,d K and compute 4.5and 4.6 to obtain d i,j,d,d,...,d K, update x i,j,y i,j,...,yk i,j =x i,jd i,j,y i,j d,...,y K and go to step 3. Step 3 Update parameter η Set η i = θη i for some θ >, increase i by and go to step. i,j dk Lemma 4.. Any global minimizer d i,j,d,d,...,d K of the problems 4.3 and 4.4satisfies the equation Q i,j µ i,j Id i,j = C i,j, 4.0 Q k µ i,j I k d k i,j = Ck, for k =,,...,K, 4. which Q i,j µ i,j I and Q k µ i,j I k for k =,,...,K are positive semi-definite µ i,j 0 and µ i,j d i,j K d k α i,j K α k = 0. This Lemma is well-known in the trust-region literature. For proof see e.g. Section 7. of [5]. Theorem 4.. If we choose α i,j K α k =, then we have 4 f ηi x i,j,y i,j,y i,j,...,yk i,j f η i x i,j,y i,j,y i,j,...,yk i,j < which is independent of i and j. Proof. If the solution of equation 4.3 and 4.4 line on to boundary of the trust-region, i.e., d i,j K d k = α i,j K α k we have q i,jd i,j,d,d,...,d K = d i,j,q i,j d i,j C i,j,d i,j d k,q k d k C k,d k,

14 50 Thai J. Math. 5 07/ P. Kabcome and T. Mouktonglang = d i,j,q i,j d i,j C i,j d i,j,q i,j d i,j d k,q k d k C k d k,q k d k d i,j, η i F x i,j H0 F x i,j I d i,j = d i,j,µ i,j d i,j d k,µ i,j d k d k, η i p k F y k i,j k H F y k i,j I k d k = µ i,j d i,j d k d i,j, η i F x i,j H0 F x i,j d i,j d k, η i p k F y k k i,j H F y k i,j = µ i,j α i,j α k d k d i,j d i,j, η i F x i,j H0 F x i,j d i,j d k, η i p k F yk k i,j H F yk i,j d k α i,j α i,j d k α k α k = In the above, the third equality follows from the equalities 4.9, 4., 4.0 and 4.. The inequality follows from the fact that η i F x i,j H 0 F x i,j and η i p k F yk i,j H k F yk i,j for k =,,...,K are positive definite or positive semidefinite. Therefore we get the last inequality. From 4.7 and 4.3 we get f ηi x i,j,y i,j,y i,j,...,yk i,j f η i x i,j,y i,j,y i,j,...,yk i,j 3 /43 3 /4 < If the solution of 4.3 and 4.4 lies in the interior of the trust-region, i.e., d i,j K d k < α i,j K α k. From Lemma 4., we know µi,j = 0 and consequently d i,j = Q i,j C i,j, d k = Q k C k for k =,,...,K which gives q i,jd i,j,d,d,...,d K = d i,j,q i,j d i,j C i,j,d i,j d k,q k d k C k,d k

15 An Interior-Point Trust-Region Algorithm for Quadratic... 5 = C,i,j,Q i,j C,i,j = C,i,j,Q i,j C,i,j Ck,Q k C k C k,q k C k = n η i X i,j X i,j. 4.5 By4.5andthemechanismofouralgorithm,weknowthat n ηi X i,j X i,j > 36 for all i and j we can rewrite 4.7 as f ηi x i,j,y i,j,y i,j,...,yk i,j f η i x i,j,y i,j,y i,j,...,yk α i,j K n η i X i,j X i,j 3 7 /43 3 /4 < 45. The proof is completed α i,j K α k 3 α k i,j 4.6 Now, weconsiderthe numberofiterationofouralgorithmthat wecan stopthe iteration when the reduction of objective function is smaller than some constant. Lemma 4.3. Let X = argmin X K qx and X = [x,y,y,,y K ] T R n n k,x K,y k K for k =,,...,K. If n η X X 6, then qx qx ϑ ϑ. 4.7 η Proof. Let Xη = argmin X K f η X. From Lemma.5 X Xη X < and from Lemma.3 we have Consider X Xη Xη X Xη X X Xη X <. 4.9 qxη qx q Xη,Xη X = F Xη,Xη X η ϑ η. 4.30

16 5 Thai J. Math. 5 07/ P. Kabcome and T. Mouktonglang The first inequality follows from the convexity of qx. The equality follows from the fact that f Xη = 0 and the last inequality follows from Lemma.6. From the inequality 4.8 and 4.9 Then, we have qx qxη = q Xη,X Xη X Xη,Q X Xη F Xη =,X Xη X Xη,ηQ X Xη η η F Xη F Xη,F Xη X Xη η η X Xη X F Xη F Xη F Xη X Xη 8η = F Xη,F Xη F Xη X Xη,F Xη X Xη 8η ϑ X Xη Xη η 8η ϑ η ϑ 8η η 4.3 where the third last inequality uses the Lemma.6 and from the fact that ϑ is always greater than. By adding the inequalities 4.30 and 4.3, then we have The proof is completed qx qx ϑ ϑ. η The above Lemma tells us that to get an ǫ solution. Let η i = θη i for some θ > and, we only need provided that the number of outer iterations i satisfies Lemma 4.4. If n ηi X X 6, then η i = η 0 θ i ϑ ϑ, 4.3 ǫ i lnϑ ϑ/ǫη lnθ f ηi X f ηi Xηi θϑ ϑ. 4.34

17 An Interior-Point Trust-Region Algorithm for Quadratic Proof. From the convexity of f ηi X and the inequality 4.8, we can show that f ηi X f ηi Xη i f η i X,X Xη i = η i QX CF X,X Xη i = η i η i ηi QX CF X,X Xη i η i F X,Xη i X η i = θ f η i X η i QX CF X,f η i XX Xη i θ F X F X,F X Xηi X θ f η i X η i QX CF X f θ F X F X F X Xηi X η i X X Xη i θ n ηi X X Xη i X X θ ϑ Xη i X X Consider θ 6 3 θ ϑ 3 θ ϑ f ηi Xηi f ηi Xη i f η i Xη i,xη i Xη i = η i QXη i CF Xη i,xη i Xη i = η i η i QXη i CF Xη i,xη i Xη i η i η i F Xη i,xη i Xη i η i = θ f η i Xηi,Xη i Xη i θ F Xη i,xη i Xη i = θ F Xη i,xη i Xη i < θϑ, 4.36 where the last equality follows from the fact that Xη i minimizes f ηi X which implies that f η i X = 0, the last inequality follows from Lemma.6. By adding inequalities 4.35 and 4.36, then we have The proof is completed f ηi X f ηi Xηi θϑ ϑ.

18 54 Thai J. Math. 5 07/ P. Kabcome and T. Mouktonglang Theorem 4. and Lemma 4.4 tell us that the steps in each inner iteration in at most steps in each inner iteration. 45θϑ ϑ 4.37 Theorem 4.5. If the initial point X 0,0 satisfies the conditions 4.9, for any ǫ > 0, our algorithm obtains the solution X which satisfies qx qx < ǫ in at most steps, here X = argmin X K qx. 45θϑ ϑln ϑ ϑ ǫη 0 lnθ 4.38 Proof. Inequality 4.33 provides us with the number of outer iterations i and Theorem 4.and Lemma 4.4 providesus with the number of steps in each iteration j. The number of iterations is at most the number of outer iterations multiply the number of inner iterations then we have the bound of number of iteration in at most steps. 45θϑ ϑ ln ϑ ϑ ǫη 0 lnθ Consider the case when the initial point X 0,0 does not satisfy condition 4.9. We can start from the analytic center of the feasible set K, since K is a bounded convex set. Let Xη 0 = argmin X K f η0 X and X = argmin X K qx. Since X 0,0 = argmin X K FX, we have f η0 X 0,0 f η0 Xη0 η 0 qx0,0 qx 4.39 and we choose η 0 / qx 0,0 qx, from Theorem 4. implies that condition 4.9 will be satisfied after at most 45 steps. 5 Implementing the Algorithm In this section we discuss the performance of our algorithm when the domain of the problem is in nonnegative orthant cones and second-order cones. These cones are all well known examples of symmetric cone. We consider multi dimension real sets. We compare our results with those of the MATLAB optimization toolbox is it called quadprog function in cases of the nonnegative orthant cones. Our algorithm can solve stochastic symmetric programming problems that include variables in second- order cone and cones of real symmetric positive semidefinite matrices. The quadprog function is unable to solve problems with these variables.

19 An Interior-Point Trust-Region Algorithm for Quadratic Problem Statement The parameter chosen for deterministic data of SQSP are shown below [ ] [ ] H 0 = c = A = B = The random variables H k,wk,t k,h k,d k, k =,,...,K. For simplicity, we consider the simple case K = 4 scenarios but the procedure can be easily extended to very large K. H H H H W W W 3 W 4 [ ] h h h 3 h 4 [ ]

20 56 Thai J. Math. 5 07/ P. Kabcome and T. Mouktonglang T T T 3 T [ ] g g g 3 g we randomly generate probability function p in 6 cases as shown below. p p p 3 p 4 case case case case case case Results of an Interior-Point Trust-Region Algorithm In order to make a comparison, the problems are described in chapter 4. Our algorithm is then implemented under MATLAB environment release 05a. The results are in table -, for case nonnegative orthant cones, we compare our results with those of the quadprog function in MATLAB optimization toolbox and our algorithm as in table in table and. In particular, we present our results of stochastic second order cone programming as in Table 3. From the numerical results example, we have the same optimal solutions with those of the quadprog function and our algorithm. The numerical results for stochastic second order cone programming, some case of problem are same optimal solutions and some case are difference because the restrictions of cones.

21 An Interior-Point Trust-Region Algorithm for Quadratic Table : The solution of SQSP when K and K are nonnegative cone by quadprog function for R n. Case Case Case3 Case4 Case5 Case6 x x x x y y y y y y y y y 3 4.5E E E E y y y y y y y y y y E00 y y Optimal value

22 58 Thai J. Math. 5 07/ P. Kabcome and T. Mouktonglang Table : The solution of SQSP when K and K are nonnegative con by an interior-point trust-region algorithm for R n. Case Case Case3 Case4 Case5 Case6 x x x x y y y y y y y y y y y y y y y y y y y y y Optimal value

23 An Interior-Point Trust-Region Algorithm for Quadratic Table 3: The solution of SQSP when K and K are second-order cone by an interior-point trust-region algorithm for SOCP. Case Case Case3 Case4 Case5 Case6 x x x x y y y y y y y y y y y y y y y y y y y y y Optimal value

24 60 Thai J. Math. 5 07/ P. Kabcome and T. Mouktonglang 6 Conclusion and Remarks In this paper, we present stochastic symmetric programming with linear function and quadratic function and finite scenarios. We can explicitly formulate the problem as a large scale deterministic programs. The complexity of our algorithm is proved to be as good as the interior-point polynomial algorithm. We have verified our performance of the algorithm on simple case study problems. Numerical results show the effectiveness of our algorithm. Acknowledgement : The authors would like to thank the Thailand Research Fund under Project RTA and Chiang Mai University, Chiang Mai, Thailand, for the financial support. References [] Y. Lu, Y. Yuan, An interior-point trust-region algorithm for general symmetric cone programming, SIAM Journal on Optimization, [] J. Faraut, A. Kornyi, Analysis on Symmetric Cones, Oxford Univ Press, Oxford, 994. [3] Y.E. Nesterov, A.S. Nemirovski, Interior-point Polynomial Algorithms in Convex Programming, SIAM Publications, SIAM, Philadelphia, USA, 994. [4] J. Renegar, A Mathematical View of Interior-Point Methods in Convex Optimization, SIAM Publications, SIAM, Philadelphia, USA, 00. [5] A.R. Conn, N.I.M. Gould, Ph.L. Toint, Trust-region Methods, SIAM Publications, SIAM, Philadelphia, USA, 000. Received February 07 Accepted 3 April 07 Thai J. Math.