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Semidefinite programs (SDPs)—some of the most useful and versatile optimization problems of the last few decades—are often pathological: the optimal values of the primal and dual problems may differ and may not be attained. Such SDPs are both theoretically interesting and often impossible to solve; yet, the pathological SDPs in the literature look strikingly similar. Based on our recent work [G. Pataki, SIAM J. Optim., 27 (2017), pp. 146-172] we characterize pathological semidefinite systems by certain excluded matrices, which are easy to spot in all published examples. Our main tool is a normal (canonical) form of a semidefinite system, which makes its pathological behavior easy to verify. The normal form is constructed in a surprisingly simple fashion, using mostly elementary row operations inherited from Gaussian elimination. The proofs are elementary and can be followed by a reader at the advanced undergraduate level. As a byproduct, we show how to transform any linear map acting on symmetric matrices into a normal form, which allows us to quickly check whether the image of the semidefinite cone under the map is closed. We can thus introduce readers to a fundamental issue in convex analysis: the linear image of a closed convex set may not be closed, and often simple conditions are available to verify the closedness, or lack of it.

In this paper we present different regularity conditions that equivalently characterize various ɛ-duality gap statements (with ɛ ≥ 0) for constrained optimization problems and their Lagrange and Fenchel-Lagrange duals in separated locally convex spaces, respectively. These regularity conditions are formulated by using epigraphs and ɛ-subdifferentials. When ɛ = 0 we rediscover recent results on stable strong and total duality and zero duality gap from the literature.

Linear programming problems in fuzzy environment have been investigated by many researchers in the recent years. Some researchers have solved these problems by using primal-dual method with linear and exponential membership functions. These membership functions are particular form of the reference functions. In this paper, we introduce a pair of primal-dual PPs in Atanassov’s intuitionistic fuzzy environment (AIFE) in which the membership and non-membership functions are taken in the form of the reference functions and prove duality results in AIFE by using an aspiration level approach with different view points, viz., optimistic, pessimistic and mixed. Since fuzzy and AIF environments cause duality gap, we propose to investigate the impact of membership functions governed by reference functions on duality gap. This is specially meaningful for fuzzy and AIF programming problems, when the primal and dual objective values may not be bounded. Finally, the duality gap obtained by the approach has been compared with the duality gap obtained by existing approaches.

We use a Fenchel duality scheme for solving control-constrained linear-quadratic optimal control problems. We derive the dual of the optimal control problem explicitly, where the control constraints are embedded in the dual objective functional, which turns out to be continuously differentiable. We specifically prove that strong duality and saddle point properties hold. We carry out numerical experiments with the discretized primal and dual formulations of the problem, for which we implement powerful existing finite-dimensional optimization techniques and associated software. We illustrate that by solving the dual of the optimal control problem, instead of the primal one, significant computational savings can be achieved. Other numerical advantages are also discussed. [PUBLICATION ABSTRACT]

The polyhedral projection problem arises in various applications. To efficiently solve the dual problem, one of the crucial issues is to safely identify zero-elements as well as the signs of nonzero elements at the optimal solution. In this paper, relying on its nonsmooth dual problem and active set techniques, we first propose a Duality-Gap-Active-Set strategy (DGASS) to effectively identify the indices of zero-elements and the signs of nonzero entries of the optimal solution. Serving as an efficient acceleration strategy, DGASS can be embedded into certain iterative methods. In particular, by applying DGASS to both the proximal gradient algorithm (PGA) and the proximal semismooth Newton algorithm (PSNA), we propose the method of PGA-DGASS and PSNA-DGASS, respectively. Global convergence and local quadratic convergence rate are discussed. We report on numerical results over both synthetic and real data sets to demonstrate the high efficiency of the two DGASS-accelerated methods.

Primal or dual strong-duality (or min-sup, inf-max duality) in nonconvex optimization is revisited in view of recent literature on the subject, establishing, in particular, new characterizations for the second case. This gives rise to a new class of quasiconvex problems having zero duality gap or closedness of images of vector mappings associated to those problems. Such conditions are described for the classes of linear fractional functions and that of quadratic ones. In addition, some applications to nonconvex quadratic optimization problems under a single inequality or equality constraint, are presented, providing new results for the fulfillment of zero duality gap or dual strong-duality.

This paper deals with the problem of linear programming with inexact data represented by real intervals. We introduce the concept of duality gap to interval linear programming. We give characterizations of strongly and weakly zero duality gap in interval linear programming and its special case where the matrix of coefficients is real. We show computational complexity of testing weakly- and strongly zero duality gap for commonly used types of interval linear programming.

We consider the problem of minimizing a sum of non-convex functions over a compact domain, subject to linear inequality and equality constraints. Approximate solutions can be found by solving a convexified version of the problem, in which each function in the objective is replaced by its convex envelope. We propose a randomized algorithm to solve the convexified problem which finds an ϵ -suboptimal solution to the original problem. With probability one, ϵ is bounded by a term proportional to the maximal number of active constraints in the problem. The bound does not depend on the number of variables in the problem or the number of terms in the objective. In contrast to previous related work, our proof is constructive, self-contained, and gives a bound that is tight.

The aim of this paper is to study an optimal control problem governed by a quasihemivariational inequality by using nonlinear Lagrangian methods. We first show the existence of solutions to the inequality problem, and then, we establish several sufficient conditions for the zero duality gap property between the optimal control problem and its nonlinear dual problem.

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.