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Comparative study of RPSALG algorithm for convex semi-infinite programming

- Auslender, Alfred, Ferrer, Albert, Goberna, Miguel, Lopez, Marco

**Authors:**Auslender, Alfred , Ferrer, Albert , Goberna, Miguel , Lopez, Marco**Date:**2014**Type:**Text , Journal article**Relation:**Computational Optimization and Applications Vol. 60, no. 1 (2014), p. 59-87**Full Text:**false**Reviewed:****Description:**The Remez penalty and smoothing algorithm (RPSALG) is a unified framework for penalty and smoothing methods for solving min-max convex semi-infinite programing problems, whose convergence was analyzed in a previous paper of three of the authors. In this paper we consider a partial implementation of RPSALG for solving ordinary convex semi-infinite programming problems. Each iteration of RPSALG involves two types of auxiliary optimization problems: the first one consists of obtaining an approximate solution of some discretized convex problem, while the second one requires to solve a non-convex optimization problem involving the parametric constraints as objective function with the parameter as variable. In this paper we tackle the latter problem with a variant of the cutting angle method called ECAM, a global optimization procedure for solving Lipschitz programming problems. We implement different variants of RPSALG which are compared with the unique publicly available SIP solver, NSIPS, on a battery of test problems.

Towards supremum-sum subdifferential calculus free of qualification conditions

- Correa, Rafael, Hantoute, Abderrahim, Lopez, Marco

**Authors:**Correa, Rafael , Hantoute, Abderrahim , Lopez, Marco**Date:**2016**Type:**Text , Journal article**Relation:**Siam Journal on Optimization Vol. 26, no. 4 (2016), p. 2219-2234**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We give a formula for the subdifferential of the sum of two convex functions where one of them is the supremum of an arbitrary family of convex functions. This is carried out under a weak assumption expressing a natural relationship between the lower semicontinuous envelopes of the data functions in the domain of the sum function. We also provide a new rule for the subdifferential of the sum of two convex functions, which uses a strategy of augmenting the involved functions. The main feature of our analysis is that no continuity-type condition is required. Our approach allows us to unify, recover, and extend different results in the recent literature.

**Authors:**Correa, Rafael , Hantoute, Abderrahim , Lopez, Marco**Date:**2016**Type:**Text , Journal article**Relation:**Siam Journal on Optimization Vol. 26, no. 4 (2016), p. 2219-2234**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We give a formula for the subdifferential of the sum of two convex functions where one of them is the supremum of an arbitrary family of convex functions. This is carried out under a weak assumption expressing a natural relationship between the lower semicontinuous envelopes of the data functions in the domain of the sum function. We also provide a new rule for the subdifferential of the sum of two convex functions, which uses a strategy of augmenting the involved functions. The main feature of our analysis is that no continuity-type condition is required. Our approach allows us to unify, recover, and extend different results in the recent literature.

Farkas-type results for vector-valued functions with applications

- Dinh, Nguyen, Goberna, Miguel, Lopez, Marco, Mo, T. H.

**Authors:**Dinh, Nguyen , Goberna, Miguel , Lopez, Marco , Mo, T. H.**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 173, no. 2 (2017), p. 357-390**Full Text:****Reviewed:****Description:**The main purpose of this paper consists of providing characterizations of the inclusion of the solution set of a given conic system posed in a real locally convex topological space into a variety of subsets of the same space defined by means of vector-valued functions. These Farkas-type results are used to derive characterizations of the weak solutions of vector optimization problems (including multiobjective and scalar ones), vector variational inequalities, and vector equilibrium problems.

**Authors:**Dinh, Nguyen , Goberna, Miguel , Lopez, Marco , Mo, T. H.**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 173, no. 2 (2017), p. 357-390**Full Text:****Reviewed:****Description:**The main purpose of this paper consists of providing characterizations of the inclusion of the solution set of a given conic system posed in a real locally convex topological space into a variety of subsets of the same space defined by means of vector-valued functions. These Farkas-type results are used to derive characterizations of the weak solutions of vector optimization problems (including multiobjective and scalar ones), vector variational inequalities, and vector equilibrium problems.

Stability of semi-infinite inequality systems involving min-type functions

- Lopez, Marco, Rubinov, Alex, Vera De Serio, Virginia

**Authors:**Lopez, Marco , Rubinov, Alex , Vera De Serio, Virginia**Date:**2005**Type:**Text , Journal article**Relation:**Numerical Functional Analysis and Optimization Vol. 26, no. 1 (2005), p. 81-112**Full Text:**false**Reviewed:****Description:**We study the stability of semi-infinite inequality systems that arise in monotonic analysis. These systems are defined by certain classes of abstract linear functions. We consider the cone R**Description:**C1**Description:**2003001420

Stability of the lower level sets of ICAR functions

- Lopez, Marco, Rubinov, Alex, Vera De Serio, Virginia

**Authors:**Lopez, Marco , Rubinov, Alex , Vera De Serio, Virginia**Date:**2005**Type:**Text , Journal article**Relation:**Numerical Functional Analysis and Optimization Vol. 26, no. 1 (2005), p. 113-127**Full Text:**false**Reviewed:****Description:**In this paper, we study the stability of the lower level set {x E R++n | f (x) ≤ 0} of a finite valued increasing convex-along-rays (ICAR) function f defined on R++n. In monotonic analysis, ICAR functions play the role of usual convex functions in classical convex analysis. We show that each ICAR function f is locally Lipschitz on int dom f and that the pointwise convergence of a sequence of ICAR functions implies its uniform convergence on each compact subset of R ++n. The latter allows us to establish stability results for ICAR functions in some sense similar to those for convex functions. Copyright © Taylor & Francis, Inc.**Description:**C1**Description:**2003001419

**Authors:**Lopez, Marco , Volle, Michel**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 390, no. 1 (2012), p. 307-312**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:**false**Reviewed:****Description:**In this paper we approach the study of the subdifferential of the closed convex hull of a function and the related integration problem without the usual assumption of epi-pointedness. The key tool is, as in Hiriart-Urruty et al. (2011) [7], the concept of ε-subdifferential. Some other assumptions which are standard in the literature are also removed.

Stationarity and Regularity of Infinite Collections of Sets. Applications to Infinitely Constrained Optimization

- Kruger, Alexander, Lopez, Marco

**Authors:**Kruger, Alexander , Lopez, Marco**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 155, no. 2 (2012), p. 390-416**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**This article continues the investigation of stationarity and regularity properties of infinite collections of sets in a Banach space started in Kruger and López (J. Optim. Theory Appl. 154(2), 2012), and is mainly focused on the application of the stationarity criteria to infinitely constrained optimization problems. We consider several settings of optimization problems which involve (explicitly or implicitly) infinite collections of sets and deduce for them necessary conditions characterizing stationarity in terms of dual space elements-normals and/or subdifferentials.

**Authors:**Kruger, Alexander , Lopez, Marco**Date:**2012**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 155, no. 2 (2012), p. 390-416**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**This article continues the investigation of stationarity and regularity properties of infinite collections of sets in a Banach space started in Kruger and López (J. Optim. Theory Appl. 154(2), 2012), and is mainly focused on the application of the stationarity criteria to infinitely constrained optimization problems. We consider several settings of optimization problems which involve (explicitly or implicitly) infinite collections of sets and deduce for them necessary conditions characterizing stationarity in terms of dual space elements-normals and/or subdifferentials.

Stability in linear optimization and related topics. A personal tour

**Authors:**Lopez, Marco**Date:**2012**Type:**Text , Journal article**Relation:**TOP Vol. 20, no. 2 (2012), p. 217-244**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:**false**Reviewed:****Description:**This paper is a kind of biased survey of the most representative and recent results on stability for the linear optimization problem. Qualitative and quantitative approaches are considered in this survey, and some infinite-dimensional extensions of the main results to more general problems are also included. In particular the paper deals with the lower/upper semicontinuity of the feasible/optimal set mappings, different types of ill-posedness, distance to ill-posedness, Lipschitz properties of these mappings under different types of perturbations, and estimates of the associated Lipschitz bounds.

A unifying approach to robust convex infinite optimization duality

- Dinh, Nguyen, Goberna, Miguel, Lopez, Marco, Volle, Michel

**Authors:**Dinh, Nguyen , Goberna, Miguel , Lopez, Marco , Volle, Michel**Date:**2017**Type:**Text , Journal article**Relation:**Journal of Optimization Theory and Applications Vol. 174, no. 3 (2017), p. 650-685**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**This paper considers an uncertain convex optimization problem, posed in a locally convex decision space with an arbitrary number of uncertain constraints. To this problem, where the uncertainty only affects the constraints, we associate a robust (pessimistic) counterpart and several dual problems. The paper provides corresponding dual variational principles for the robust counterpart in terms of the closed convexity of different associated cones.

Weaker conditions for subdifferential calculus of convex functions

- Correa, Rafael, Hantoute, Abderrahim, Lopez, Marco

**Authors:**Correa, Rafael , Hantoute, Abderrahim , Lopez, Marco**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Functional Analysis Vol. 271, no. 5 (2016), p. 1177-1212**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**In this paper we establish new rules for the calculus of the subdifferential mapping of the sum of two convex functions. Our results are established under conditions which are at an intermediate level of generality among those leading to the Hiriart-Urruty and Phelps formula (Hiriart-Urruty and Phelps, 1993 [15]), involving the approximate subdifferential, and the stronger assumption used in the well-known Moreau-Rockafellar formula (Rockafellar 1970, [23]; Moreau 1966, [20]), which only uses the exact subdifferential. We give an application to derive asymptotic optimality conditions for convex optimization.**Description:**In this paper we establish new rules for the calculus of the subdifferential mapping of the sum of two convex functions. Our results are established under conditions which are at an intermediate level of generality among those leading to the Hiriart-Urruty and Phelps formula (Hiriart-Urruty and Phelps, 1993 [15]), involving the approximate subdifferential, and the stronger assumption used in the well-known Moreau-Rockafellar formula (Rockafellar 1970, [23]; Moreau 1966, [20]), which only uses the exact subdifferential. We give an application to derive asymptotic optimality conditions for convex optimization. (C) 2016 Elsevier Inc. All rights reserved.

Best approximate solutions of inconsistent linear inequality systems

- Goberna, Miguel, Hiriart-Urruty, Jean-Baptiste, Lopez, Marco

**Authors:**Goberna, Miguel , Hiriart-Urruty, Jean-Baptiste , Lopez, Marco**Date:**2018**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 46, no. 2 (2018), p. 271-284**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This paper is intended to characterize three types of best approximate solutions for inconsistent linear inequality systems with an arbitrary number of constraints. It also gives conditions guaranteeing the existence of best uniform solutions and discusses potential applications.

**Authors:**Goberna, Miguel , Hiriart-Urruty, Jean-Baptiste , Lopez, Marco**Date:**2018**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 46, no. 2 (2018), p. 271-284**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This paper is intended to characterize three types of best approximate solutions for inconsistent linear inequality systems with an arbitrary number of constraints. It also gives conditions guaranteeing the existence of best uniform solutions and discusses potential applications.

Convexity and closedness in stable robust duality

- Dinh, Nguyen, Goberna, Miguel, Lopez, Marco, Volle, Michel

**Authors:**Dinh, Nguyen , Goberna, Miguel , Lopez, Marco , Volle, Michel**Date:**2019**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 13, no. 2 (2019), p. 325-339**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper deals with optimization problems with uncertain constraints and linear perturbations of the objective function, which are associated with given families of perturbation functions whose dual variable depends on the uncertainty parameters. More in detail, the paper provides characterizations of stable strong robust duality and stable robust duality under convexity and closedness assumptions. The paper also reviews the classical Fenchel duality of the sum of two functions by considering a suitable family of perturbation functions.

**Authors:**Dinh, Nguyen , Goberna, Miguel , Lopez, Marco , Volle, Michel**Date:**2019**Type:**Text , Journal article**Relation:**Optimization Letters Vol. 13, no. 2 (2019), p. 325-339**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper deals with optimization problems with uncertain constraints and linear perturbations of the objective function, which are associated with given families of perturbation functions whose dual variable depends on the uncertainty parameters. More in detail, the paper provides characterizations of stable strong robust duality and stable robust duality under convexity and closedness assumptions. The paper also reviews the classical Fenchel duality of the sum of two functions by considering a suitable family of perturbation functions.

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