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.     

Subdifferentiability and the Duality Gap

We point out a connection between sensitivity analysis and the fundamental theorem of linear programming by characterizing when a linear programming problem has no duality gap. The main result is that the value function is subdifferentiable at the primal constraint if and only if there exists an optimal dual solution and there is no duality gap. To illustrate the subtlety of the condition, we extend Kretschmer's gap example to construct (as the value function of a linear programming problem) a convex function which is subdifferentiable at a point but is not continuous there. We also apply the theorem to the continuum version of the assignment model.     

Robust conjugate duality for convex optimization under uncertainty with application to data classification

In this paper we present a robust conjugate duality theory for convex programming problems in the face of data uncertainty within the framework of robust optimization, extending the powerful conjugate duality technique. We first establish robust strong duality between an uncertain primal parameterized convex programming model problem and its uncertain conjugate dual by proving strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem under a regularity condition. This regularity condition is not only sufficient for robust duality but also necessary for it whenever robust duality holds for every linear perturbation of the objective function of the primal model problem. More importantly, we show that robust strong duality always holds for partially finite convex programming problems under scenario data uncertainty and that the optimistic counterpart of the dual is a tractable finite dimensional problem. As an application, we also derive a robust conjugate duality theorem for support vector machines which are a class of important convex optimization models for classifying two labelled data sets. The support vector machine has emerged as a powerful modelling tool for machine learning problems of data classification that arise in many areas of application in information and computer sciences.     

Solutions to quadratic minimization problems with box and integer constraints

This paper presents a canonical duality theory for solving quadratic minimization problems subjected to either box or integer constraints. Results show that under Gao and Strang’s general global optimality condition, these well-known nonconvex and discrete problems can be converted into smooth concave maximization dual problems over closed convex feasible spaces without duality gap, and can be solved by well-developed optimization methods. Both existence and uniqueness of these canonical dual solutions are presented. Based on a second-order canonical dual perturbation, the discrete integer programming problem is equivalent to a continuous unconstrained Lipschitzian optimization problem, which can be solved by certain deterministic technique. Particularly, an analytical solution is obtained under certain condition. A fourth-order canonical dual perturbation algorithm is presented and applications are illustrated. Finally, implication of the canonical duality theory for the popular semi-definite programming method is revealed.     

Duality Theorems for Separable Convex Programming Without Qualifications

In the research of mathematical programming, duality theorems are essential and important elements. Recently, Lagrange duality theorems for separable convex programming have been studied. Tseng proves that there is no duality gap in Lagrange duality for separable convex programming without any qualifications. In other words, although the infimum value of the primal problem equals to the supremum value of the Lagrange dual problem, Lagrange multiplier does not always exist. Jeyakumar and Li prove that Lagrange multiplier always exists without any qualifications for separable sublinear programming. Furthermore, Jeyakumar and Li introduce a necessary and sufficient constraint qualification for Lagrange duality theorem for separable convex programming. However, separable convex constraints do not always satisfy the constraint qualification, that is, Lagrange duality does not always hold for separable convex programming. In this paper, we study duality theorems for separable convex programming without any qualifications. We show that a separable convex inequality system always satisfies the closed cone constraint qualification for quasiconvex programming and investigate a Lagrange-type duality theorem for separable convex programming. In addition, we introduce a duality theorem and a necessary and sufficient optimality condition for a separable convex programming problem, whose constraints do not satisfy the Slater condition.     

Comment on a note on duality gaps in linear programming over convex sets

In Ref. 1, Soyster has given a rather complicated proof of the absence of a duality gap, under a certain interiority condition, for a variant of a pair of optimization problems introduced by Ben-Israel, Charnes, and Kortanek (Ref. 2). A proof can be given directly (and under weaker conditions) by a simple application of a Lagrange multiplier theorem on convex programming in abstract spaces (Ref. 3).     

Dual Semidefinite Programs Without Duality Gaps for a Class of Convex Minimax Programs

In this paper, we introduce a new dual program, which is representable as a semidefinite linear programming problem, for a primal convex minimax programming problem, and we show that there is no duality gap between the primal and the dual whenever the functions involved are sum-of-squares convex polynomials. Under a suitable constraint qualification, we derive strong duality results for this class of minimax problems. Consequently, we present applications of our results to robust sum-of-squares convex programming problems under data uncertainty and to minimax fractional programming problems with sum-of-squares convex polynomials. We obtain these results by first establishing sum-of-squares polynomial representations of non-negativity of a convex max function over a system of sum-of-squares convex constraints. The new class of sum-of-squares convex polynomials is an important subclass of convex polynomials and it includes convex quadratic functions and separable convex polynomials. The sum-of-squares convexity of polynomials can numerically be checked by solving semidefinite programming problems whereas numerically verifying convexity of polynomials is generally very hard.     

A projection method for the uncapacitated facility location problem

Several algorithms already exist for solving the uncapacitated facility location problem. The most efficient are based upon the solution of the strong linear programming relaxation. The dual of this relaxation has a condensed form which consists of minimizing a certain piecewise linear convex function. This paper presents a new method for solving the uncapacitated facility location problem based upon the exact solution of the condensed dual via orthogonal projections. The amount of work per iteration is of the same order as that of a simplex iteration for a linear program inm variables and constraints, wherem is the number of clients. For comparison, the underlying linear programming dual hasmn + m + n variables andmn +n constraints, wheren is the number of potential locations for the facilities. The method is flexible as it can handle side constraints. In particular, when there is a duality gap, the linear programming formulation can be strengthened by adding cuts. Numerical results for some classical test problems are included.     

A generalized Lagrangian function and multiplier method

As is well known, a saddle point for the Lagrangian function, if it exists, provides a solution to a convex programming problem; then, the values of the optimal primal and dual objective functions are equal. However, these results are not valid for nonconvex problems.In this paper, several results are presented on the theory of the generalized Lagrangian function, extended from the classical Lagrangian and the generalized duality program. Theoretical results for convex problems also hold for nonconvex problems by extension of the Lagrangian function. The concept of supporting hypersurfaces is useful to add a geometric interpretation to computational algorithms. This provides a basis to develop a new algorithm.     

On the dual of linear inverse problems

In linear inverse problems considered in this paper a vector with positive components is to be selected from a feasible set defined by linear constraints. The selection rule involves minimization of a certain function which is a measure of distance from a priori guess. Csiszar made an axiomatic approach towards defining a family of functions, we call it α-divergence, that can serve as logically consistent selection rules. In this paper we present an explicit and perfect dual of the resulting convex programming problem, prove the corresponding duality theorem and optimality criteria, and make some suggestions on an algorithmic solution.     

Robust duality for generalized convex programming problems under data uncertainty

In this paper we present a robust duality theory for generalized convex programming problems in the face of data uncertainty within the framework of robust optimization. We establish robust strong duality for an uncertain nonlinear programming primal problem and its uncertain Lagrangian dual by showing strong duality between the deterministic counterparts: robust counterpart of the primal model and the optimistic counterpart of its dual problem. A robust strong duality theorem is given whenever the Lagrangian function is convex. We provide classes of uncertain non-convex programming problems for which robust strong duality holds under a constraint qualification. In particular, we show that robust strong duality is guaranteed for non-convex quadratic programming problems with a single quadratic constraint with the spectral norm uncertainty under a generalized Slater condition. Numerical examples are given to illustrate the nature of robust duality for uncertain nonlinear programming problems. We further show that robust duality continues to hold under a weakened convexity condition.     

Duality for a class of nondifferentiable mathematical programming problems

Two dual problems are proposed for the minimax problem: minimize max_y?Yφ(x, y), subject to g(x) ? 0. A duality theorem is established for each dual problem. It is revealed that these problems are intimately related to a class of nondifferentiable programming problems.     

Optimality and duality in nondifferentiable multiobjective mathematical programming involving higher order (F,α,ρ,d)-type I functions

In this paper, a nondifferentiable multiobjective programming problem is considered where every component of objective and constraint functions contain a term involving the support function of a compact convex set. A new class of higher order (F,α,ρ,d)-type I function is introduced. Necessary optimality conditions and the duality theorems for Wolfe and unified higher order dual problems are established. Several known results can be deduced as special cases.     

On reduction of duality gap in quadratic knapsack problems

We investigate in this paper the duality gap between quadratic knapsack problem and its Lagrangian dual or semidefinite programming relaxation. We characterize the duality gap by a distance measure from set {0, 1} ⁿ to certain polyhedral set and demonstrate that the duality gap can be reduced by an amount proportional to the square of the distance. We further discuss how to compute the distance measure via cell enumeration method and to derive the corresponding improved upper bound of the problem.     

A reduced gradient method for quadratic programs with quadratic constraints and lp-constrained lp-approximation problems

A dual algorithm is developed for solving a general class of nonlinear programs that properly contains all convex quadratic programs with quadratic constraints and l_p-constrained l_p-approximation problems. The general dual program to be solved has essentially linear constraints but the objective function is nondifferentiable when certain variables equal zero. Modifications to the reduced gradient method for linearly constrained problems are presented that overcome the numerical difficulties associated with the nondifferentiable objective function. These modifications permit ‘blocks’ of variables to move to and away from zero on certain iterations even though the objective function is nondifferentiable at points having a block of variables equal to zero.     

A note on strong duality in convex semidefinite optimization: necessary and sufficient conditions

A strong duality which states that the optimal values of the primal convex problem and its Lagrangian dual problem are equal (i.e. zero duality gap) and the dual problem attains its maximum is a corner stone in convex optimization. In particular it plays a major role in the numerical solution as well as the application of convex semidefinite optimization. The strong duality requires a technical condition known as a constraint qualification (CQ). Several CQs which are sufficient for strong duality have been given in the literature. In this note we present new necessary and sufficient CQs for the strong duality in convex semidefinite optimization. These CQs are shown to be sharper forms of the strong conical hull intersection property (CHIP) of the intersecting sets of constraints which has played a critical role in other areas of convex optimization such as constrained approximation and error bounds. Research was partially supported by the Australian Research Council. The author is grateful to the referees for their helpful comments     

Higher order duality in multiobjective fractional programming with support functions

In this paper a new class of higher order (F,ρ,σ)-type I functions for a multiobjective programming problem is introduced, which subsumes several known studied classes. Higher order Mond-Weir and Schaible type dual programs are formulated for a nondifferentiable multiobjective fractional programming problem where the objective functions and the constraints contain support functions of compact convex sets in Rn. Weak and strong duality results are studied in both the cases assuming the involved functions to be higher order (F,ρ,σ)-type I. A number of previously studied problems appear as special cases.     

Partially finite convex programming,Part II: Explicit lattice models

In Part I of this work we derived a duality theorem for partially finite convex programs, problems for which the standard Slater condition fails almost invariably. Our result depended on a constraint qualification involving the notion ofquasi relative interior. The derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function: the differentiability of various convex functions in lattices was considered at the end of Part I. In Part II we shall apply our results to a number of more concrete problems, including variants of semi-infinite linear programming,L ¹ approximation, constrained approximation and interpolation, spectral estimation, semi-infinite transportation problems and the generalized market area problem of Lowe and Hurter (1976). As in Part I, we shall use lattice notation extensively, but, as we illustrated there, in concrete examples lattice-theoretic ideas can be avoided, if preferred, by direct calculation.     

Solving dual problems using a coevolutionary optimization algorithm

In solving certain optimization problems, the corresponding Lagrangian dual problem is often solved simply because in these problems the dual problem is easier to solve than the original primal problem. Another reason for their solution is the implication of the weak duality theorem which suggests that under certain conditions the optimal dual function value is smaller than or equal to the optimal primal objective value. The dual problem is a special case of a bilevel programming problem involving Lagrange multipliers as upper-level variables and decision variables as lower-level variables. Another interesting aspect of dual problems is that both lower and upper-level optimization problems involve only box constraints and no other equality of inequality constraints. In this paper, we propose a coevolutionary dual optimization (CEDO) algorithm for co-evolving two populations—one involving Lagrange multipliers and other involving decision variables—to find the dual solution. On 11 test problems taken from the optimization literature, we demonstrate the efficacy of CEDO algorithm by comparing it with a couple of nested smooth and nonsmooth algorithms and a couple of previously suggested coevolutionary algorithms. The performance of CEDO algorithm is also compared with two classical methods involving nonsmooth (bundle) optimization methods. As a by-product, we analyze the test problems to find their associated duality gap and classify them into three categories having zero, finite or infinite duality gaps. The development of a coevolutionary approach, revealing the presence or absence of duality gap in a number of commonly-used test problems, and efficacy of the proposed coevolutionary algorithm compared to usual nested smooth and nonsmooth algorithms and other existing coevolutionary approaches remain as the hallmark of the current study.     

Robust linear semi-infinite programming duality under uncertainty

In this paper, we propose a duality theory for semi-infinite 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-infinite 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 establish that the closed-convex robust moment cone condition in the case of constraint-wise uncertainty is in fact necessary and sufficient for robust duality. In other words, the 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 affinely parameterized data uncertainty, we establish that robust duality is easily satisfied under a Slater type constraint qualification. Consequently, we derive robust forms of the Farkas lemma for systems of uncertain semi-infinite linear inequalities.