Preliminaries on Linear Semi-infinite Optimization

Goberna, Miguel Á.; López, Marco A.

doi:10.1007/978-1-4899-8044-1_1

Cited by 95 publications

(213 citation statements)

References 189 publications

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“…From the linear SIP strong duality theorem, inf(RSP ) = max(DRSP ) whenever K is closed (see, e.g., [15,Chapter 8]). If M is closed and convex, then K = co M = M is closed and so…”

Section: Theorem 1 If the Robust Moment Cone M Is Closed And Convex mentioning

confidence: 99%

“…(a t (v t ) ; b t (w t )) is continuous and so, the set f(a t (v t ); b t (w t )); (t; u t ) 2 gph Ug is compact. As we are assuming that the Slater condition holds, K is closed by [15,Theorem 5.3 (ii)]. Finally, the closedness of K and [15, (8.5)-(8.6)] imply inf(RSP ) = max(DRSP ):…”

Section: Robust Semi-in…nite Farkas'lemmamentioning

confidence: 99%

“…We further show that the closed-convex robust moment cone condition in the case of constraint uncertainty is in fact necessary and su¢ cient for robust duality in the sense that robust moment cone is closed and convex if and only if robust duality holds for every linear objective function of the program. We also derive strong duality between the robust counterpart (RSP) and its standard (or Haar) dual problem [15] in terms of a robust characteristic cone, illustrating the link between the Haar dual and the optimistic dual.…”

Section: Introductionmentioning

confidence: 96%

“…Duality theory has played a key role in the study of semi-in…nite programming [13,15,16,22] which traditionally assumes perfect information (that is, accurate values for the input quantities or system parameters), despite the reality that such precise knowledge is rarely available in practice for realworld optimization problems. The data of real-world optimization problems more often than not are uncertain (that is, they are not known exactly at the time of the decision) due to estimation errors, prediction errors or lack of information [3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

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Robust linear semi-infinite programming duality under uncertainty

Goberna

Jeyakumar

et al. 2013

Math. Program.

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In this paper, we propose a duality theory for semi-in…nite linear programming problems under uncertainty in the constraint functions, the objective function, or both, within the framework of robust optimization. We present robust duality by establishing strong duality between the robust counterpart of an uncertain semi-in…nite linear program and the optimistic counterpart of its uncertain Lagrangian dual. We show that robust duality holds whenever a robust moment cone is closed and convex. We then show that the closed-convex robust moment cone condition in the case of constraint-wise uncertainty is in fact necessary and su¢ cient for robust duality in the sense that robust moment cone is closed and convex if and only if robust duality holds for every linear objective function of the program. In the case of uncertain problems with a¢ nely parameterized data uncertainty, we establish that robust duality is easily satis…ed under a Slater type constraint quali…cation. Consequently, we derive robust forms of the Farkas lemma for systems of uncertain semi-in…nite linear inequalities.

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“…From the linear SIP strong duality theorem, inf(RSP ) = max(DRSP ) whenever K is closed (see, e.g., [15,Chapter 8]). If M is closed and convex, then K = co M = M is closed and so…”

Section: Theorem 1 If the Robust Moment Cone M Is Closed And Convex mentioning

confidence: 99%

Section: Robust Semi-in…nite Farkas'lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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Robust linear semi-infinite programming duality under uncertainty

Goberna

Jeyakumar

et al. 2013

Math. Program.

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“…Denoting by v P (a, b, c) the optimal value of P and by v D (a, b, c) the optimal value of D, the duality gap of (a, b, c) is Observe that v P and v D are positively homogeneous, so that the duality gap function g : Θ → R + ∪ {+∞} is positively homogeneous too. Some works, as [4] and [10], define the duality gap as 0 when the primal and dual problems have both the same infinite value (e.g. when one of them is inconsistent and the other is unbounded).…”

mentioning

confidence: 99%

Stability of the Duality Gap in Linear Optimization

Goberna

Ridolfi

Serio

2017

Set-Valued Var. Anal

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In this paper we consider the duality gap function g that measures the difference between the optimal values of the primal problem and of the dual problem in linear programming and in linear semi-infinite programming. We analyze its behavior when the data defining these problems may be perturbed, considering seven different scenarios. In particular we find some stability results by proving that, under mild conditions, either the duality gap of the perturbed problems is zero or +∞ around the given data, or g has an infinite jump at it. We also give conditions guaranteeing that those data providing a finite duality gap are limits of sequences of data providing zero duality gap for sufficiently small perturbations, which is a generic result.

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Characterizations of Robust and Stable Duality for Linearly Perturbed Uncertain Optimization Problems

Dinh¹,

Goberna²,

López³

et al. 2020

Springer Proceedings in Mathematics &Amp; Statistics

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We introduce a robust optimization model consisting in a family of perturbation functions giving rise to certain pairs of dual optimization problems in which the dual variable depends on the uncertainty parameter. The interest of our approach is illustrated by some examples, including uncertain conic optimization and infinite optimization via discretization. The main results characterize desirable robust duality relations (as robust zero-duality gap) by formulas involving the epsilon-minima or the epsilon-subdifferentials of the objective function. The two extreme cases, namely, the usual perturbational duality (without uncertainty), and the duality for the supremum of functions (duality parameter vanishing) are analyzed in detail.

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Preliminaries on Linear Semi-infinite Optimization

Cited by 95 publications

References 189 publications

Robust linear semi-infinite programming duality under uncertainty

Robust linear semi-infinite programming duality under uncertainty

Stability of the Duality Gap in Linear Optimization

Characterizations of Robust and Stable Duality for Linearly Perturbed Uncertain Optimization Problems

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