Strong duality for standard convex programs

Kortanek, K. O.; Yu, Guolin; Zhang, Qinghong

doi:10.1007/s00186-021-00761-x

Cited by 2 publications

(5 citation statements)

References 21 publications

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“…Notice that the dual formulation (RDP) is closely related to the dual problem proposed for the standard convex program by Kortanek et al [18], but it is not a direct consequence of this result. Problems (FDP) and (RDP) are equivalent but written in different forms, with the problem (RDP) having a compact form that resembles the form of the Lagrange dual problem (2.4).…”

Section: Discussionmentioning

confidence: 99%

“…In [18], the authors used a polynomial ring approach developed in [12,14] to formulate a strong dual for a standard convex optimization program. The aim of this section is to show how the strong dual problems (DP) and (FDP) considered in this paper for the copositive problem (COP) can be reformulated in terms of this approach.…”

Section: Reformulations Of Problems (Dp) and (Fdp) Using A Polynomial...mentioning

confidence: 99%

“…This made it possible to obtain dual characterizations of optimality that do not require any CQ. In the recent paper [18], this approach was used to develop a strong duality for standard convex programming problems.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, motivated by the approach presented in [18,19] for conic problems and using the results from [15,16], we deduce several new strong dual formulations for copositive problems without relying on any CQs. The main aim of the paper is to study properties of the proposed strong duals and provide a detailed comparison between them.…”

Section: Introductionmentioning

confidence: 99%

“…Other dual formulations, (EDP) and (FDP), of the linear copositive problem are studied in Section 5. Reformulations of the duals (DP) and (FDP) using the polynomial ring approach developed in [12,14,18] are described in Section 6. The paper is ended with some conclusions.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Linear copositive programming: Strong dual formulations and their properties

2023

Commu. Optim. Theory

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In Copositive Programming, a cost function is optimized over a cone of matrices that are positive semidefinite in the non-negative ortant. Being a fairly new field of research, Copositive Programming has already gained popularity. Duality theory is a rich and powerful area of convex optimization, which is central to understanding sensitivity analysis and infeasibility issues as well as to development of numerical methods. In this paper, we continue our recent research on dual formulations for linear Copositive Programming. The dual problems obtained in the paper satisfy the strong duality relations and do not require any additional regularity assumptions such as constraint qualifications. Different dual formulations have their own special properties, the corresponding feasible sets are described in different ways, so they can have an independent application in practice.

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Section: Discussionmentioning

confidence: 99%

Section: Reformulations Of Problems (Dp) and (Fdp) Using A Polynomial...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Linear copositive programming: Strong dual formulations and their properties

2023

Commu. Optim. Theory

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Exploring constraint qualification-free optimality conditions for linear second-order cone programming

Kostyukova,

Tchemisova

2024

Int. J. Optim. Control, Theor. Appl. (IJOCTA)

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Linear second-order cone programming (SOCP) deals with optimization problems characterized by a linear objective function and a feasible region defined by linear equalities and second-order cone constraints. These constraints involve the norm of a linear combination of variables, enabling the representation of a wide range of convex sets. The SOCP serves as a potent tool for addressing optimization challenges across engineering, finance, machine learning, and various other domains. In this paper, we introduce new optimality conditions tailored for {SOCP} problems. These conditions have the form of two optimality criteria that are obtained without the requirement of any constraint qualifications and are defined explicitly. The first criterion utilizes the concept of immobile indices of constraints. The second criterion, without relying explicitly on immobile indices, introduces a special finite vector set for assessing optimality. To demonstrate the effectiveness of these criteria, we present two illustrative examples highlighting their applicability. We compare the obtained criteria with other known optimality conditions and show the advantage of the former ones.

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Strong duality for standard convex programs

Cited by 2 publications

References 21 publications

Linear copositive programming: Strong dual formulations and their properties

Linear copositive programming: Strong dual formulations and their properties

Exploring constraint qualification-free optimality conditions for linear second-order cone programming

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