Zero Duality Gap Research Articles

On duality gap with polynomial multipliers for polynomial optimization problems

In this paper, we examine a zero-duality gap with polynomial multipliers between a non-convex polynomial program and its associated generalized Lagrangian dual problem. To this end, we introduce a sum of squares certificate of non-negativity, abbreviated as (SC), that allows us to verify the non-negativity of a polynomial over a feasible set of a system of given polynomials. With the help of (SC), we show that the zero-duality gap with polynomial multipliers coincides with the convergent Lasserre hierarchy of semidefinite programming relaxations. This zero-duality gap with polynomial multipliers reduces to the classical zero-duality gap with constant multipliers when considering it in the setting of SOS-convex polynomial programs or quadratic programs with the hidden convexity. We also show that, in the framework of truncated polynomial systems, the zero-duality gap with polynomial multipliers can be checked via the classical Karush–Kuhn–Tucker condition under an additional convexity condition.

Zero duality gap conditions via abstract convexity

Using tools provided by the theory of abstract convexity, we extend conditions for zero duality gap to the context of non-convex and nonsmooth optimization. Mimicking the classical setting, an abstract convex function is the upper envelope of a family of abstract affine functions (being conventional vertical translations of the abstract linear functions). We establish new conditions for zero duality gap under no topological assumptions on the space of abstract linear functions. In particular, we prove that the zero duality gap property can be fully characterized in terms of an inclusion involving (abstract) -subdifferentials. This result is new even for the classical convex setting. Endowing the space of abstract linear functions with the topology of pointwise convergence, we extend several fundamental facts of functional/convex analysis. This includes (i) the classical Banach–Alaoglu–Bourbaki theorem (ii) the subdifferential sum rule, and (iii) a constraint qualification for zero duality gap which extends a fact established by Borwein, Burachik and Yao (2014) for the conventional convex case. As an application, we show with a specific example how our results can be exploited to show zero duality for a family of non-convex, non-differentiable problems.

Moreau Envelope of Supremum Functions with Applications to Infinite and Stochastic Programming

In this paper, we investigate the Moreau envelope of the supremum of a family of convex, proper, and lower semicontinuous functions. Under mild assumptions, we prove that the Moreau envelope of a supremum is the supremum of Moreau envelopes, which allows us to approximate possibly nonsmooth supremum functions by smooth functions that are also the suprema of functions. Consequently, we propose and study approximated optimization problems from infinite and stochastic programming for which we obtain zero-duality gap results and optimality conditions without the verification of constraint qualification conditions.

The Zero Duality Gap Property for an Optimal Control Problem Governed by a Multivalued Hemivariational Inequality

We show in this work the zero duality gap property for an optimal control problem governed by a multivalued hemivariational inequality with unbounded constraint set. Based on the existence of solutions to the inequality, we establish several sufficient conditions for the zero duality gap property between the optimal control problem and its nonlinear dual problem by using nonlinear Lagrangian methods. Moreover, we obtain a convergence result for the optimal control problem governed by a perturbed multivalued hemivariational inequality.

Duality for constrained robust sum optimization problems

Given an infinite family of extended real-valued functions \(f_{i}\), \(i\in I,\) and a family \({\mathcal {H}}\) of nonempty finite subsets of I, the \({\mathcal {H}}\)-partial robust sum of \(f_{i}\), \(i\in I,\) is the supremum, for \(J\in {\mathcal {H}},\) of the finite sums \(\sum _{j\in J}f_{j}\). These infinite sums arise in a natural way in location problems as well as in functional approximation problems, and include as particular cases the well-known sup function and the so-called robust sum function, corresponding to the set \( {\mathcal {H}}\) of all nonempty finite subsets of I, whose unconstrained minimization was analyzed in previous papers of three of the authors (https://doi.org/10.1007/s11228-019-00515-2 and https://doi.org/10.1007/s00245-019-09596-9). In this paper, we provide ordinary and stable zero duality gap and strong duality theorems for the minimization of a given \({\mathcal {H}}\)-partial robust sum under constraints, as well as closedness and convex criteria for the formulas on the subdifferential of the sup-function.

New regularity conditions and Fenchel dualities for DC optimization problems involving composite functions

We consider the following DC composite optimization problem (P): where and are proper convex functionals defined on locally convex Hausdorff topological vector spaces X and Y respectively, and h is a proper K-convex mapping from X to Y. By using the properties of the epigraph of the conjugate functions, we introduce some new regularity conditions and obtain complete characterizations for weak / strong / stable Fenchel dualities and for zero / stable zero duality gap properties of problem (P).

Min\u2013sup-type zero duality gap properties for DC composite optimization problem

In this paper, we present min–sup-type zero duality gap properties for DC composite optimization problem with conic constraints. Using properties of the subdifferentials of involved functions, we introduce some new constraint qualifications. Under these new constraint qualifications, we provide necessary and/or sufficient conditions for the min–sup-type stable zero duality to hold.

Modified Courant-Beltrami penalty function and a duality gap for invex optimization problem

In this paper, we modified a Courant-Beltrami penalty function method for constrained optimization problem to study a duality for convex nonlinear mathematical programming problems. Karush-Kuhn-Tucker (KKT) optimality conditions for the penalized problem has been used to derived KKT multiplier based on the imposed additional hypotheses on the constraint function g. A zero-duality gap between an optimization problem constituted by invex functions with respect to the same function η and their Lagrangian dual problems has also been established. The examples have been provided to illustrate and proved the result for the broader class of convex functions, termed invex functions.

Optimal Precoder Designs for Sum-Utility Maximization in SWIPT-Enabled Multi-User MIMO Cognitive Radio Networks

In this paper, we propose a generalized framework that combines the cognitive radio (CR) techniques for spectrum sharing and the simultaneous wireless information and power transfer (SWIPT) for energy harvesting (EH) in the conventional multi-user MIMO (MuMIMO) channels, which leads to an MuMIMO-CR-SWIPT network. In this system, we have one secondary base-station (S-BS) that supports multiple secondary information decoding (S-ID) and secondary EH (S-EH) users simultaneously under the condition that interference power that affects the primary ID (P-ID) receivers should stay below a certain threshold. The goal of the paper is to develop a generalized precoder design that maximizes the sum-utility cost function under the transmit power constraint at the S-BS, and the EH constraint at each S-EH user, and the interference power constraint at each P-ID user. Therefore, the previous studies for the CR and SWIPT systems are casted as particular solutions of the proposed framework. The problem is inherently non-convex and even the weighted minimum mean squared error (WMMSE) transformation does not resolve the non-convexity of the original problem. To tackle the problem, we find a solution from the dual optimization via sub-gradient ellipsoid method based on the observation that the WMMSE transformation raises zero-duality gap between the primal and the dual problems. We also propose a simplified algorithm for the case of a single S-ID user, which is shown to achieve the global optimum. Finally, we demonstrate the optimality and efficiency of the proposed algorithms through numerical simulation results.

Minimax Duality for MIMO Interference Networks

A minimax duality for a Gaussian mutual information expression was introduced by Yu. An interesting observation is the relationship between cost constraints on the transmit covariances and noise covariances in the dual problem via Lagrangian multipliers. We introduce a minimax duality for general MIMO interference networks, where noise and transmit covariances are optimized subject to linear conic constraints. We observe a fully symmetric relationship between the solutions of both networks, where the roles of the optimization variables and Lagrangian multipliers are inverted. The least favorable noise covariance itself provides a Lagrangian multiplier for the linear conic constraint on the transmit covariance in the dual network, while the transmit covariance provides a Lagrangian multiplier for the constraint on the interference plus noise covariance in the dual network. The degrees of freedom available for optimization are constituted by linear subspaces, where the orthogonal subspaces induce the constraints in the dual network. For the proof of our duality we make use of the existing polite water-filling network duality and as a by-product we are able to show that maximization problems in MIMO interference networks have a zero-duality gap for a special formulation of the dual function. Our minimax duality unifies and extends several results, including the original minimax duality and other known network dualities. New results and applications are MIMO transmission strategies that manage and handle uncertainty due to unknown inter-cell interference and information theoretic proofs concerning cooperation in networks and optimality of proper signaling.

A Lagrangian Dual Approach to the Single-Source Localization Problem

The single-source localization problem (SSLP), which is nonconvex by its nature, appears in several important multidisciplinary fields such as signal processing and the global positioning system. In this paper, we cast SSLP as a Euclidean distance embedding problem and study a Lagrangian dual approach. It is proved that the Lagrangian dual problem must have an optimal solution under the generalized Slater condition. We provide a sufficient condition for the zero-duality gap and establish the equivalence between the Lagrangian dual approach and the existing Generalized Trust-Region Subproblem (GTRS) approach studied by Beck et al. [“Exact and Approximate Solutions of Source Localization Problems,” IEEE Trans. Signal Process., vol. 56, pp. 1770-1778, 2008]. We also reveal new implications of the assumptions made by the GTRS approach. Moreover, the Lagrangian dual approach has a straightforward extension to the multiple-source localization problem. Numerical simulations demonstrate that the Lagrangian dual approach can produce localization of similar quality as the GTRS and can significantly outperform the well-known semidefinite programming solver for the multiple source localization problem on the tested cases.

Zero duality gap in surrogate constraint optimization: A concise review of models

Surrogate constraint relaxation was proposed in the 1960s as an alternative to the Lagrangian relaxation for solving difficult optimization problems. The duality gap in the surrogate relaxation is always as good as the duality gap in the Lagrangian relaxation. Over the years researchers have proposed procedures to reduce the gap in the surrogate constraint. Our aim is to review models that close the surrogate duality gap. Five research streams that provide procedures with zero duality gap are identified and discussed. In each research stream, we will review major results, discuss limitations, and suggest possible future research opportunities. In addition, relationships between models if they exist, are also discussed.

Zero Duality Gap for Convex Programs: A Generalization of the Clark–Duffin Theorem

This article uses classical notions of convex analysis over Euclidean spaces, like Gale & Klee’s boundary rays and asymptotes of a convex set, or the inner aperture directions defined by Larman and Brondsted for the same class of sets, to provide a generalization of the Clark–Duffin Theorem.

Resource Allocation for Interweave and Underlay CRs Under Probability-of-Interference Constraints

Efficient design of cognitive radios (CRs) calls for secondary users implementing adaptive resource allocation schemes that exploit knowledge of the channel state information (CSI), while at the same time limiting interference to the primary system. This paper introduces stochastic resource allocation algorithms for both interweave (also known as overlay) and underlay cognitive radio paradigms. The algorithms are designed to maximize the weighted sum-rate of orthogonally transmitting secondary users under average-power and probabilistic interference constraints. The latter are formulated either as short- or as long-term constraints, and guarantee that the probability of secondary transmissions interfering with primary receivers stays below a certain pre-specified level. When the resultant optimization problem is non-convex, it exhibits zero-duality gap and thus, due to a favorable structure in the dual domain, it can be solved efficiently. The optimal schemes leverage CSI of the primary and secondary networks, as well as the Lagrange multipliers associated with the constraints. Analysis and simulated tests confirm the merits of the novel algorithms in: i) accommodating time-varying settings through stochastic approximation iterations; and ii) coping with imperfect CSI.

Stable zero duality gaps in convex programming: Complete dual characterisations with applications to semidefinite programs

The zero duality gap that underpins the duality theory is one of the central ingredients in optimisation. In convex programming, it means that the optimal values of a given convex program and its associated dual program are equal. It allows, in particular, the development of efficient numerical schemes. However, the zero duality gap property does not always hold even for finite-dimensional problems and it frequently fails for problems with non-polyhedral constraints such as the ones in semidefinite programming problems. Over the years, various criteria have been developed ensuring zero duality gaps for convex programming problems. In the present work, we take a broader view of the zero duality gap property by allowing it to hold for each choice of linear perturbation of the objective function of the given problem. Globalising the property in this way permits us to obtain complete geometric dual characterisations of a stable zero duality gap in terms of epigraphs and conjugate functions. For convex semidefinite programs, we establish necessary and sufficient dual conditions for stable zero duality gaps, as well as for a universal zero duality gap in the sense that the zero duality gap property holds for each choice of constraint right-hand side and convex objective function. Zero duality gap results for second-order cone programming problems are also given. Our approach makes use of elegant conjugate analysis and Fenchel's duality.

On Zero Duality Gap and the Farkas Lemma for Conic Programming

Recently S. A. Clark published an interesting duality result in linear conic programming dealing with a convex cone that is not closed in which the usual (algebraic) dual problem is replaced by a topological dual with the aim of having zero duality gap under certain usual hypotheses met in mathematical finance. We present some examples to show that an extra condition is needed to reach a conclusion; this supplementary condition is also provided. We also give counterexamples for three results on hedging prices and simple proofs for two known solvability results (see Propositions 4.1 and 4.2).

A Lagrange penalty reformulation method for constrained optimization

In this paper a constrained optimization problem is transformed into an equivalent one in terms of an auxiliary penalty function. A Lagrange function method is then applied to this transformed problem. Zero duality gap and exact penalty results are obtained without any coercivity assumption on either the objective function or constraint functions.

An Optimal Alternative Theorem and Applications to Mathematical Programming

Given a closed convex cone P with nonempty interior in a locally convex vector space, and a set $$A \subset Y $$ , we provide various equivalences to the implication $$A \cap (-{\rm int}\,P) = \emptyset \Longrightarrow {\rm co}(A)\cap (-{\rm int}\, P) = \emptyset, $$ among them, to the pointedness of cone(A + int P). This allows us to establish an optimal alternative theorem, suitable for vector optimization problems. In addition, we present an optimal alternative theorem which characterizes two-dimensional spaces in the sense that it is valid if, and only if, the space is at most two-dimensional. Applications to characterizing weakly efficient solutions through scalarization; the zero (Lagrangian) duality gap; the Fritz---John optimality conditions for a class of nonconvex nonsmooth minimization problems, are also presented.

The Zero Duality Gap Property and Lower Semicontinuity of the Perturbation Function

We examine the validity of the zero duality gap properties for two important dual schemes: a generalized augmented Lagrangian dual scheme and a nonlinear Lagrange-type dual scheme. The necessary and sufficient conditions for the zero duality gap property to hold are established in terms of the lower semicontinuity of the perturbation functions.

Asymptotic Strong Duality for Bounded Integer Programming: A Logarithmic-Exponential Dual Formulation

A logarithmic-exponential dual formulation is proposed in this paper for bounded integer programming problems. This new dual formulation possesses an asymptotic strong duality property and guarantees the identification of an optimal solution of the primal problem. These prominent features are achieved by exploring a novel nonlinear Lagrangian function, deriving an asymptotic zero duality gap, investigating the unimodality of the associated dual function and ensuring the primal feasibility of optimal solutions in the dual formulation. One other feature of the logarithmicexponential dual formulation is that no actual dual search is needed when parameters are set above certain threshold-values.