J.B. LASSERRE AND E.S. ZERON

Transcription

1 L -NORMS, LOG-BARRIERS AND CRAMER TRANSFORM IN OPTIMIZATION J.B. LASSERRE AND E.S. ZERON Abstract. We show that the Lalace aroxiation of a sureu by L -nors has interesting consequences in otiization. For instance, the logarithic barrier functions (LBF) of a rial convex roble P and its dual P aear naturally when using this sile aroxiation technique for the value function g of P or its Legendre-Fenchel conjugate g. In addition, iniizing the LBF of the dual P is just evaluating the Craer transfor of the Lalace aroxiation of g. Finally, this technique erits to soeties define an exlicit dual roble P in cases when the Legendre-Fenchel conjugate g cannot be derived exlicitly fro its definition. 1. Introduction Let f : R and ω : R be a air of continuous aings defined on the convex cone R n. Consider the function g : R R given by the forula: (1.1) y g(y) := su f(x) : ω(x) y, x. x For each fixed y R, couting g(y) is solving the otiization roble (1.2) P : su f(x) : ω(x) y, x, x and g is called the value function associated with P. The value function g rovides a systeatic way to generate a dual roble P via its Legendre- Fenchel conjugate denoted g : R R. The concave version of g is defined by (1.3) g () := inf y R y g(y), and is finite on soe doain D R. Then the dual roble reduces to (1.4) (1.5) P : g(y) = inf y g () = inf y g () : D = inf R + su f(x) + (y ω(x) Matheatics Subject Classification. 90C25 90C26. Key words and hrases. otiization; Logarithic Barrier Function; Legendre-Fenchel and Craer transfors. 1

2 2 J.B. LASSERRE AND E.S. ZERON Weak duality ilies g(y) g(y), that is, (1.6) (1.7) g(y) = su inf R + inf f(x) + (y ω(x) R + su f(x) + (y ω(x) = g(y), and the equality g(y) = g(y) holds true under soe convexity assution. However, in general g cannot be obtained exlicitly fro its definition (1.3), and for dual ethods to solve P, the inner axiization in (1.7) ust be done nuerically for each fixed. A notable excetion is the conic otiization roble where f and ω are both linear aings, for which one ay obtain an exlicit dual (1.4). Of course, alternative exlicit duals have been roosed but they involve both rial (x) and dual () variables. In articular, the Wolfe [14] and Mond-Weir [11] duals even allow to consider weakened notions of convexity like e.g. seudo- or quasi-convexity. For a nice exosition and related references on this toic, the interested reader is referred to Mond [12] and the references therein. Contribution. Our contribution is to show that the sile and wellknown Lalace aroxiation of a sureu via a converging sequence of L -nors has interesting consequences in otiization, for both rial and dual robles P and P. Recall that the celebrated logarithic barrier function (LBF in short) associated with a convex otiization roble P as in (1.2), or with its dual P, is well-known for its good technical roerties like e.g. the selfconcordance (which holds in soe cases) which exlains its good nuerical efficiency; see e.g. [5, 6]. But the LBF is only one articular choice aong any other interior enalty functions! We first show that the LBF of the rial roble P (with araeter ) aears naturally by using the sile and well-known Lalace aroxiation of a sureu via L -nors, alied to the inner infiu in (1.6). It is a bit surising to obtain an efficient ethod in this way. Indeed, the inner infiu in (1.6) (which is exactly equal to zero for a feasible x) is relaced with its naive Lalace aroxiation by L -nors, and to the best of our knowledge, the efficiency of this aroxiation has not been roved or even tested nuerically! Siilarly, when using the sae Lalace L -nor aroxiation technique for the infiu in the definition (1.3) of the conjugate function g, we obtain a function φ : R R which: (a) deends on an integer araeter and (b), is valid on the relative interior ri D of soe doain D R. In doing so for conic otiization robles, the set D is just the feasible set of the (known) exlicit dual roble P, and φ is (u to a constant) the LBF with araeter, associated with P. So again, for conic rogras, the sile Lalace aroxiation of a sureu by L -nors erits to retrieve the LBF of the dual roble P! Interestingly, the function y in φ (; y) is nothing less than the Craer transfor of the Lalace aroxiation e f L (Ω(y)), where Ω(y)

3 L -NORMS AND CRAMER TRANSFORM IN OPTIMIZATION 3 is the feasible set of roble P and L is the usual nor associated with the Lebesgue sace L. Analogies between the Lalace and Fenchel transfors via exonentials and logariths in the Craer transfor have been already exlored in other contexts, in order to establish nice arallels between otiization and robability via a change of algebra; see e.g. Bacelli et al. [1], Maslov [10], Lasserre [9], and the any references therein. In robability, the Craer transfor of a robability easure has also been used to rovide exact asytotics of soe integrals as well as to derive large deviation rinciles. For a nice survey on this toic the interested reader is referred to Piterbarg and Falatov [13]. In addition, an interesting feature of this Lalace aroxiation technique is to rovides us with a systeatic way to obtain a dual roble (1.4) in cases when g cannot be obtained exlicitly fro its definition (1.3). Naely, in a nuber of cases and in contrast with g, the function φ (; y) obtained by using the Lalace aroxiation of the conjugate function g by L -nors, can be couted in closed-for exlicitly. Exales of such situations are briefly discussed. In the general case, φ is of the for φ(; y) + ψ(, ) where: for every ri D fixed, the function ψ(, ) 0 as ; and for each fixed, the function ψ(, ) is a barrier for the doain D. This yields to consider the otiization roble P : in φ(; y) : D as a natural dual of P, and for which φ is an associated barrier function with araeter. If g is concave then strong duality holds. 2. Main result We need soe interediary helful results before stating our ain result Soe reliinary results. Let L q () be the usual Lebesgue sace of integrable functions defined on a Borel-easurable set R n, and h L q () (or soeties h q ) be the associated nor h L q () = h q := ( 1/q h(x) dx) q. To ake the aer self-contained we rove the following known result. Lea 1. Let R n be any Borel-easurable set, and h L q () for soe given q 1, so that h L q () <. Then: li h L () = h := ess su h(x). Proof. Notice that ay be an unbounded set. Suose that h q < for soe given q > 1, and define Λ to be the essential sureu of h in

4 4 J.B. LASSERRE AND E.S. ZERON. The result is trivial when Λ = 0, so we assue that Λ (0, ). Then ( ess su h(x) = Λ = li h/λ L q () [ = li [ ( h(x) Λ Λ ) q/λ ) q dx] 1/ 1/ (2.1) li h(x) dx] = li h L (). It is also obvious that Λ li h when Λ =. On the other hand, suose that the essential sureu Λ of h in is finite. Given an arbitrary araeter ɛ > 0, there exists a bounded subset B with ositive finite Lebesgue easure (B) (0, ) such that h(x) > Λ ɛ for every x B. Then li h L () li h L (B) li (B)1/ (Λ ɛ) = Λ ɛ. Therefore, since ɛ is arbitrary, cobining the revius identity with (2.1) yields the desired result li h L () = Λ. In the sae way, assue that the essential sureu of h in is infinite. Given an arbitrary natural nuber N N, there exists a bounded subset B with ositive finite Lebesgue easure (B) (0, ) such that h(x) > N for every x B. Then li h L () li h L (B) li (B)1/ N = N. Therefore, since N is arbitrary, cobining the revius identity with (2.1) yields the desired result li h L () = Λ =. Next we also need the following interediate result. Lea 2. For every N let U R n be soe oen subset, and let h : U R be a sequence of functions indexed by the araeter N. Suose that h converges ointwise to a function h defined on an oen subset U of R n. Then: li inf h (x) inf h(x), x U x U rovided that the liit in the left side of the equation exists in the extended interval [, ). Proof. Suose that the infiniu of h on U is equal to. For every N R there is a oint x U such that h(x) < N, and so there is also an index 0 such that x U and h (x) < N for every > 0. Hence the infiniu of h on U is strictly less than N, and so li inf h (x) = = inf h(x), x U x U because N R is arbitrary. On the other hand, assue that the infiniu of h on U is equal to R. For every ɛ > 0 there is a oint x U such that h(x) < +ɛ, and so there is also an index 0 such that x U and

5 L -NORMS AND CRAMER TRANSFORM IN OPTIMIZATION 5 h (x) < +ɛ for every > 0. Since the infiniu of h on U is strictly less than +ɛ and ɛ > 0 is arbitrary, li inf h (x) = inf h(x). x U x U 2.2. L -nor aroxiations for the rial. Let us go back to roble P in (1.1) where R n is a convex cone, and let Z := R +. Let R n be the dual cone associated with, and let x (x) be the universal logarithic barrier function associated with the convex cone, that is, ( ) x (x) := ln e x y dy, x int, where int denotes the interior of. See e.g. Güller [4] and Güler and Tuncel [5]. Next, let H R n be the set H := x R n : ω(x) < y; x int. Recalling that P is a axiization roble, the LBF associated with the (rial) roble P, and with araeter N, is the function ψ : H R defined by: (2.2) x ψ (x) := f(x) + 1 (x) + ln(y ω(x)) j, The LBF in convex rograing dates back to Frisch [3] and becae widely known later in Fiacco and McCorick [2]. For ore details and a discussion, see e.g. den Hertog [6, Chater 2]. It is well-known that under soe convexity assutions, and if g(y) <, (2.3) g(y) = li su x ψ (x) : x H, and the sequence of iniizers (x ) N H of ψ converges to a iniizer of P. We next rovide a sile rationale that exlains how the LBF naturally aears to solve roble P. Observe that (1.6) can be rewritten as (2.4) g(y) = su x R n inf f(x) + (,µ) Z (y ω(x)) + µ x, and in the above equation, the inner iniization inf f(x) + (,µ) Z (y ω(x)) + µ x, (whose value is 0 if ω(x) y and x ), ay be rewritten as f(x) su (ω(x) y) µ x. (,µ) Z

6 6 J.B. LASSERRE AND E.S. ZERON But for fixed x R n and y R, one ay aroxiate the above sureu via L -nors. Indeed, if y R and x H, then for every N, e (ω(x) y) µ x dµ d <, Z and so e (ω(x) y) µ x L (Z ) <, N. Therefore, by Lea 1, su (ω(x) y) µ x = li ln (ω(x) y) µ x (,µ) Z e L (Z ) = li ( ) 1 ln e (ω(x) y) µ x dµ d Z 1 (x) 1 ln(y ω(x)) j li if x H = + otherwise. Next, observe that for each N, (x) = n (x) because is a cone, and so (2.4) now reads 1 (2.5) g(y) = su f(x) + li (x) + ln(y ω(x)) j x H. A direct alication of Lea 2 to (2.5) yields g(y) li su 1 f(x) + li (x) + x H = li su x ψ (x) : x H,, ln(y ω(x)) j and so (2.3) states that in fact one also has the reverse inequality, that is, in (2.5) one ay interchange the su and li oerators. In other words, the LBF ψ aears naturally when one aroxiates inf f(x) + (,µ) Z (y ω(x)) + µ x, (whose value is exactly zero if ω(x) y and x ), by the quantity 1 (x) + ln(y ω(x)) j, which coes fro the Lalace aroxiation of a su by L -nors. For instance, in Linear rograing where = R n +, x c x and x ω(x) = Ax for soe vector c R n and soe atrix A R n, g(y) = li su c x + 1 n ln(y Ax) j + ln(x i ) : x H. x i=1

7 L -NORMS AND CRAMER TRANSFORM IN OPTIMIZATION L -nor aroxiations for the dual. We now use the sae aroxiation technique via L -nors to either retrieve the known dual P when it is exlicit, or to rovide an exlicit dual roble P in cases where g cannot be obtained exlicitly fro its definition (1.3). Recall that if g is concave, uer sei-continuous, and bounded fro above by soe linear function, then by Legendre-Fenchel duality, (2.6) (2.7) g(y) = inf g () := inf y y g (), where y g(y). One can exress g in ters of the definition (1.1) of g and the involved continuous transforations f and ω. Naely, g () = su g(y) y y (2.8) = su su f(x) y y, ω(x) y su f(x) ω(x) if 0, (2.9) = + otherwise. Therefore the doain of definition D R of g is given by: (2.10) D := R : 0, su f(x) ω(x) <, with relative interior denoted by ri D. Observe that D is convex because g is convex and roer on D. Theore 3. Let g and g be as in (1.1) and (2.7), resectively. Assue that g is concave, uer sei-continuous, and bounded fro above by soe linear function. Suose that the relative interior ri D is not ety and for every ri D there exists an exonent q 1 such that (2.11) e f(x) ω(x) <. L q () Then: (2.12) g(y) = li inf y + ln ri D Proof. In view of (2.8) (2.13) g () = [ ln su y e f(x) ω(x) L () ] su e y+f(x) if D ω(x) y + otherwise. ln( j )

8 8 J.B. LASSERRE AND E.S. ZERON Hyothesis (2.11) and Lea 1 allow us to relace the sureu in (2.13) by the liit of the L -nors as. Naely, ( ) 1/ g () = li ln e y+f(x) dy dx (2.14) = li ln ( = li ln ω(x) y 1/ e dx) ω(x)+f(x) e f(x) ω(x) L () ln( j ) Hence fro (2.14), equation (2.6) can be rewritten as follows: g(y) = inf y g () = (2.15) = inf ri D li ri D li inf ri D y + ln y + ln e f(x) ω(x) e f(x) ω(x) L () L (). ln( j ) ln( j ) ln( j ) where we have alied Lea 2 in order to interchange the inf and li oerators. Notice that the ters between the brackets are the functions h () of Lea 2. On the other hand, given y R, let Θ(y) := (x, z) R + : ω(x) + z y R n+, so that whenever ri D, e f(x) L (Θ(y)) e f(x)+ (y ω(x) z) L (Θ(y)) e y = e y e f(x) ω(x) z L ( R + ) e f(x) ω(x) L () ( j ) 1/. By hyothesis (2.11), given ri D fixed, the L -nor ter in the last above identity is finite for soe large enough. Therefore, by definition (1.1) and Lea 1, one obtains g(y) = su ln e f(x) e = li ln f(x) li (x,z) Θ(y) inf ri D y + ln e f(x) ω(x) L () L (Θ(y)) which cobined with (2.15) yields the desired result (2.12)., ln( j ),

9 L -NORMS AND CRAMER TRANSFORM IN OPTIMIZATION 9 Reark 1. Condition (2.11) can be easily checked in articular cases. For instance let x ω(x) := Ax for soe atrix A R n. If := R n + and x f(x) := c x + ln x d for soe c and d in R n with d 0. The notation x d stands for the onoial n xd k k. Then f is concave and e f(x) ω(x) dx = e (c A ) x x d dx R n + = R n + n Γ(1 + d k ) (A k c k) 1+ d k <, whenever ri D := R : > 0, A > c and N. If = R and x f(x) := x Qx + c x for soe c R n and a syetric (strictly) ositive definite atrix Q R n n, then e f(x) ω(x) dx = e x Qx e (c A ) x dx <, R n R n whenever ri D := : > 0 and N. Consider next the following functions for every N : (2.16) φ (; y) := y + ln e f(x) ω(x) L () defined on soe doain of R, and (2.17) y g (y) := inf φ (; y) : ri D ln( j ) defined on R. Recall that the Craer transfor (denoted C) alied to an integrable function u : R R, is the Legendre-Fenchel transfor (denoted F) of the logarith of the Lalace transfor (denoted L) of u, i.e., u C(u) = F ln L (u). The Craer transfor is natural in the sense that the logarith of the Lalace transfor is always a convex function. For our urose, we will consider the concave version of the Fenchel transfor (2.18) û [F(û)]() = inf y y + û(y), for û : R R convex, so that û is concave. We clai that: Theore 4. The function y g (y) defined in (2.17) is the Craer transfor of the function (2.19) y g (y) := e f(x) dx = e f, L (Ω(y)) Ω(y) where Ω(y) := x : ω(x) y R n.

10 10 J.B. LASSERRE AND E.S. ZERON Proof. The result follows fro the definition of the Craer transfor C. Hence Therefore, g C( g ) := F ln L ( g ) y C( g )(y) = inf y + [ln L( g )](). [L( g )]() = e y g (y) dy = y R [ ] = e y e f(x) dx dy y R, ω(x) y [ ] = e f(x) e y dy dx y ω(x) [ ] = e f(x) ω(x) 1 dx j = e f(x) ω(x) 1. L () j [ln L( g )]() = ln e f(x) ω(x) ln( j ). L () On the other hand, recall the definition of g (y) given in (2.17)-(2.16), g (y) = inf y + ln e f(x) ω(x) ln( j ). ri D L () Thus, with F as in (2.18) and D := z : z D, we obtain the desired result : g (y) = inf y + ln e f(x) ω(x) ln( j ) ri D L () = inf y + [ln L( g )]() ri D = inf z y + [ln L( g )](z) z ri D = [F ln L( g )](y) = [C( g )](y), For linear rograing, this result was already obatined in [8, 9]. Exale 1. (Linear Prograing) In this case set the cone = R n + and the functions f(x) := c x and ω(x) = Ax for soe vector c R n and atrix

11 L -NORMS AND CRAMER TRANSFORM IN OPTIMIZATION 11 A R n. We easily have that = L () e f(x) ω(x) e (c A ) x dx = n 1 A k c k for every N and each in the relative interior ri D of the set (2.20) D = R : A c, 0. Hence fro (2.16) φ (; y) = y + 1 e ln f(x) ω(x) L () = y n ln(a k c k) ln( j ) ln( j ) +n = ln, One easily recognizes (u to the constant (+n)[ln ]/) the LBF with araeter, of the dual roble: P : in y : A c, 0. Exale 2. (The general conic roble) Consider the conic otiization roble in c x : ω x y, x, x for soe convex cone R n, soe vector c R n, and soe linear aing ω : R n R with adjoint aing ω : R R n. We easily have that e c x ω x e = (c ω ) x = L () L () = e (c ω ) x dx = n e (c ω ) x dx, because is a cone. Clai (2.16) reads φ (; y) = y + 1 e ln c x ω x L () (2.21) = y + ψ(ω c) ln j +n ln( j ) ln, where ψ : R n R is the so-called universal LBF associated with the dual cone, and with doain ri D, where (2.22) D = R : ω c, 0. See e.g. Güller [4] and Güler and Tuncel [5]. In φ (and u to a constant), one easily recognizes the LBF with araeter, of the dual roble: P : in y : ω c, 0.

12 12 J.B. LASSERRE AND E.S. ZERON Exale 3. (Quadratic rograing: non conic forulation) Consider syetric ositive seidefinite atrixes Q [j] R n n and vectors c [j] R n for j = 0, 1,...,. The notation Q 0 (res. Q 0) stands for Q is ositive seidefinite (res. strictly ositive definite). Let := R n, f(x) := x Q [0] x 2c [0] x and let ω : Rn R have entries ω j (x) := x Q [j] x+2c [j] x for every j = 1,...,. For R with > 0, define the real syetric atrix Q [] R n n and vector c [] R n : so that Q [] := Q [0] + j Q [j] and c [] := c [0] + e f(x) ω(x) L () j c [j], = ex ( x Q [] x 2 c [] x) dx = π ex ( c n/2 [] Q 1 [] c ) [] det ( ) <, Q [] whenever N and Q [] 0. Therefore (2.23) in 0, Q 0 φ (; y) = y + 1 e ln c x ω x L () ax = y + c [] Q 1 [] c [] ln ( ) det Q [] 2 ln j ln π ln + n ln 2, ln( j ) on the doain of definition ri D := : > 0, Q 0. Again, in equation (2.23) one easily recognizes (u to a constant) the LBF with araeter, of the dual roble P : x Q [0] x 2c [0] x ( j x ) Q [j] x+2c [j] x y j = in 0,Q 0 y + ax x Q [] x 2c [] x) = in y + c [] Q 1 [] c [] : 0, Q [] 0 where we have used the fact that x = Q 1 [] c [] R n is the unique otial solution to the inner axiization roble in the second equation above. If Q 0 0 and Q j 0, j = 1,...,, then ri D := : > 0 because Q 0 whenever > 0; in this case P is a convex otiization roble and there is no duality ga between P and P.,

13 L -NORMS AND CRAMER TRANSFORM IN OPTIMIZATION An exlicit dual. In Exales 1, 2, and 3, the function φ defined in (2.16) can be decoosed into a su of the for: (2.24) φ (; y) = h 1 (; y) + h 2 (; ) where h 1 is indeendent of the araeter. Moreover, if h 2 (; ) < for soe > 0 fixed, the ter h 2 (; ) converges to zero when. One ay also verify that h 1 () = y g (), D, where g is the Legendre-Fenchel conjugate in (2.7). In fact, the above revious decoosition is ore general and can be deduced fro soe sile facts. Recall the roble P given in (1.1), y g(y) := su f(x) : ω(x) y, x, x for a convex cone R n and continuous aings f and ω. Assue that g is concave, uer sei-continuous, and bounded fro above by soe affine function, so that Legendre-Fenchel duality yields g(y) = inf g () := inf y y g (), where y g(y), and where the doain of g is the set D R in (2.10). Lea 5. Suose that ri D is not ety and for every ri D there exists an exonent q 1 such that (2.11) holds. Then: (2.25) h 1 (; y) := li φ (; y) = y g (), ri D. Proof. Let ri D be fixed. The given hyothesis and Lea 1 ily that e li ln f(x) ω(x) = su f(x) ω(x) = L () = su f(x) y = su g(y) y = g (), y y su, ω(x) y and so (2.25) follows fro the definition (2.16) of φ. As a consequence we obtain: Corollary 6. Let D be as in (2.10), φ as in (2.16) and let h 1 (; y) be as in (2.25). Then the otiization roble (2.26) P : in h 1 (; y) : ri D. is a dual of P. Moreover, if g is concave, uer-seicontinuous and bounded above by soe affine function, then strong duality holds.

14 14 J.B. LASSERRE AND E.S. ZERON Proof. By Lea 5, h 1 (; y) = y g () for all ri D. And so if in P (res. ax P) denotes the otial value of P (res. P), one has in P = in y g () : ri D g(y) = ax P, where we have used that g () = su z g(z) z g(y) y. Finally, if g is concave, uer sei-continuous and bounded above by soe affine function, then g is convex. Therefore, as a convex function is continuous on its doain D (which is convex) in P = in h 1 (; y) : ri D that is, strong duality holds. = in y g () : ri D = in y g () : D = g(y), In a nuber of cases, the L -nor aroxiation of g can be obtained exlicitly as a function of, whereas g itself cannot be obtained exlicitly fro (1.3). In this situation one obtains an exlicit LBF φ with araeter, for soe dual P of P, and soeties an exlicit dual roble P. Indeed if φ is known exlicitly, one ay soeties get its ointwise liit h 1 (, y) in (2.25), in closed for, and so P is defined exlicitly by (2.26). With fixed, couting φ (; y) reduces to coute the integral over a convex cone of an exonential of soe function araetrized by and. Soeties this can be done with the hel of soe known transfors like e.g. the Lalace or Weierstrass transfors, as illustrated below. Linear aings and Lalace transfor. Let ω : R n R be a linear aing, with ω(x) = Ax for soe real atrix A R n, and let = R n +. Then ln e f(x) ω(x) L () = 1 ( ) ln e (A ) x e f(x) dx = 1 ln (L[e f ](A ) That is, the L -nor aroxiation is the logarith of the Lalace transfor of the function e f, evaluated at the oint A R n. So if in roble P, the objective function f is such that e f has an exlicit Lalace transfor, then one obtains an exlicit exression for the LBF φ (; y) defined in (2.16). For instance if f(x) = c x + ln q(x) for soe vector c R n and soe olynoial q R[x], ositive on the feasible set of P, write q(x) = α N n q α x α, ).

15 L -NORMS AND CRAMER TRANSFORM IN OPTIMIZATION 15 for finitely any non zero coefficients (q α ), and where the notation x α stand for the onoial x α 1 1 xαn n. Then ln ( L[e f ](A ) ) can be couted in closed-for since we have: ( ) ln L[e f ](A ) where α x α = n i=1 α i x α i i = ln ( α N n q α e (c A ) x x α dx ( ) = n ln + ln q α 1 α x α n, α N n i=1 (A c) i. Of course the above exression can becoe quite colicated, esecially for large values of. But it is exlicit in the variables ( i ). If the function x ln q(x) is concave, then Corollary 6 alies. Siilarly if f is linear, i.e. x f(x) = c x for soe vector c R n, then ( ) ln e f(x) ω(x) L () = 1 ln L[e ω(x) ](c) ), and so if the function x e ω(x) has an exlicit Lalace transfor then so does the L -nor aroxiation, and again, φ is obtained in closed for. Exale 4. As a sile illustrative exale, consider the otiization roble: P : su x c x + n b k ln x k : Ax y, x 0, x R n, for soe given atrix A R n and vectors b, c R n and y R. We suose that b 0, so that c x + ln(x b ) is concave, and in which case P is a convex rogra. Notice that with = R n +, su c x + ln(x b ) Ax : x R n, x x whenever lies in D = R : A < c, 0. We have < e c x Ax x b L () = = n e (c A ) x x b dx Γ(1+ b k ) (A k c k) 1+ b k <,

16 16 J.B. LASSERRE AND E.S. ZERON whenever N and > 0 in R satisfies A > c. Next, φ in (2.16) reads φ (; y) = y 1 e ln c x Ax x b n ln( k ) L () n [ ] ln Γ(1+ = bk ) y + b k ln(a k c k) n ln(a k c k) ln j + n ln. Stirling s aroxiation Γ(1 + t) (t/e) t 2πt for real nubers t 1 allow us to calculate the liit when goes to infinity [ ] ln Γ(1+ b k ) li b k ln = li b bk k ln b k ln e Lea 5 ilies that y g () = li φ (; y) = y = b k ln(e 1 b k ). n [ A ] b k ln k c k e 1. b k And so the function φ is the LBF with araeter, of the dual roble n [ A P : inf ] y b k ln k c k e 1 : A c, 0. b k In articular, by Corollary 6, strong duality holds, i.e., ax P = in P. References [1] F. Bacelli, G. Cohen, J. Olsder and J.P Quadrat, Syncronization and Linearity, Wiley, New York, [2] A.V. Fiaco, G.P. McCorick, Nonlinear Prograing, Sequential Unconstrained Miniization Techniques, John Wiley & Sons, New York, [3] K.R. Frisch, The logarithic otential ethod of convex rograing, Meorandu, Institute of Econoics, Oslo, Norway, [4] O. Güller, Barrier functions in interior oint ethods, Math. Oer. Res. 21 (1996), [5] O. Güller and L. Tuncel, Characterization of the barrier araeter of hoogeneous convex cones, Math. Progr. 81 (1998), [6] D. den Hertog, Interior Point Aroach to Linear, Quadratic and Convex Prograing, Kluwer, Dordrecht, [7] J.B. Hiriart-Urruty, Conditions for global otiality, in: Handbook of Global Otiization, R. Horst and P. Pardalos (Eds.), Kluwer, Dordrecht (1995), [8] J.B. Lasserre, Why the logarithic barrier function in convex and linear rograing, Oer. Res. Letters 27 (2000), [9] J.B. Lasserre, Linear and Integer Prograing Versus Linear Integration and Counting, Sringer, New York, [10] V.P. Maslov, Méthodes Oératorielles, Editions Mir, Moscou 1973, Traduction Fran caise, 1987.

17 L -NORMS AND CRAMER TRANSFORM IN OPTIMIZATION 17 [11] B. Mond and T. Weir, Generalized convexity and higher order duality, J. Math. Sci ( ), [12] B. Mond, Mond-Weir duality, in C.E.M. Pearce and E. Hunt (Eds.), Otiization: Structures and Alications, Sringer, Dordrecht (2009), [13] V.I. Piterbarg, V.R. Fatalov, The Lalace ethod for robability easures on Banach saces, Russian Math. Surveys 50 (1995), [14] F. Wolfe, A duality theore in nonlinear rograing, Quart. Al. Math. 19 (1961), LAAS-CNRS and Institute of Matheatics, University of Toulouse, LAAS, 7 avenue du Colonel Roche, Toulouse Cédex 4,France E-ail address: lasserre@laas.fr Deto. Mateaticas, CINVESTAV-IPN, Ado. Postal , Mexico, D.F , Mexico E-ail address: eszeron@ath.cinvestav.x