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.     

On zero duality gap in nonconvex quadratic programming problems

We present in this paper new sufficient conditions for verifying zero duality gap in nonconvex quadratically/linearly constrained quadratic programs (QP). Based on saddle point condition and conic duality theorem, we first derive a sufficient condition for the zero duality gap between a quadratically constrained QP and its Lagrangian dual or SDP relaxation. We then use a distance measure to characterize the duality gap for nonconvex QP with linear constraints. We show that this distance can be computed via cell enumeration technique in discrete geometry. Finally, we revisit two sufficient optimality conditions in the literature for two classes of nonconvex QPs and show that these conditions actually imply zero duality gap.     

Tightening a copositive relaxation for standard quadratic optimization problems

We focus in this paper the problem of improving the semidefinite programming (SDP) relaxations for the standard quadratic optimization problem (standard QP in short) that concerns with minimizing a quadratic form over a simplex. We first analyze the duality gap between the standard QP and one of its SDP relaxations known as “strengthened Shor’s relaxation”. To estimate the duality gap, we utilize the duality information of the SDP relaxation to construct a graph G ^?. The estimation can be then reduced to a two-phase problem of enumerating first all the minimal vertex covers of G ^? and solving next a family of second-order cone programming problems. When there is a nonzero duality gap, this duality gap estimation can lead to a strictly tighter lower bound than the strengthened Shor’s SDP bound. With the duality gap estimation improving scheme, we develop further a heuristic algorithm for obtaining a good approximate solution for standard QP.     

A scale elasticity measure for directional distance function and its dual: Theory and DEA estimation

In this paper we focus on scale elasticity measure based on directional distance function for multi-output–multi-input technologies, explore its fundamental properties and show its equivalence with the input oriented and output oriented scale elasticity measures. We also establish duality relationship between the scale elasticity measure based on the directional distance function with scale elasticity measure based on the profit function. Finally, we discuss the estimation issues of the scale elasticity based on the directional distance function via the DEA estimator.     

On duality gap in binary quadratic programming

We investigate in this paper the duality gap between the binary quadratic optimization problem and its semidefinite programming relaxation. We show that the duality gap can be underestimated by ${\xi_{r+1}\delta^2}$ , where ?? is the distance between {?1, 1} ⁿ and certain affine subspace, and ?? _r+1 is the smallest positive eigenvalue of a perturbed matrix. We also establish the connection between the computation of ?? and the cell enumeration of hyperplane arrangement in discrete geometry.     

On the existence of positive radial solutions for a certain class of elliptic BVPs

The aim of this paper is to answer the question, when a certain BVP of elliptic type possesses positive radial solutions. We develop duality and variational principles for this problem. Our approach enables the approximation of solutions and gives a measure of a duality gap between primal and dual functional for minimizing sequences.     

On the existence of multiple solutions for a nonlocal BVP with vector-valued response

The existence of positive solutions for a nonlocal boundary-value problem with vector-valued response is investigated. We develop duality and variational principles for this problem. Our variational approach enables us to approximate solutions and give a measure of a duality gap between the primal and dual functional for minimizing sequences.     

Duality Gaps in Stochastic Integer Programming

In this note, we explore the implications of a result that suggests that the duality gap caused by a Lagrangian relaxation of the nonanticipativity constraints in a stochastic mixed integer (binary) program diminishes as the number of scenarios increases. By way of an example, we illustrate that this is not the case. In general, the duality gap remains bounded away from zero.     

On the existence of bounded positive solutions for a class of singular BVPs

We investigate the existence and properties of solutions to a second-order singular ODE. We base ourselves on the variational approach, which enables the approximation of solutions and gives a measure of a duality gap between primal and dual functional for minimizing sequences.     

Starting-point strategies for an infeasible potential reduction method

We present two strategies for choosing a “hot” starting-point in the context of an infeasible potential reduction (PR) method for convex quadratic programming. The basic idea of both strategies is to select a preliminary point and to suitably scale it in order to obtain a starting point such that its nonnegative entries are sufficiently bounded away from zero, and the ratio between the duality gap and a suitable measure of the infeasibility is small. One of the two strategies is naturally suggested by the convergence theory of the PR method; the other has been devised to reduce the initial values of the duality gap and the infeasibility measure, with the objective of decreasing the number of PR iterations. Numerical experiments show that the second strategy generally performs better than the first, and both outperform a starting-point strategy based on the affine-scaling step.     

On The Existence of a Countable Set of Positive Solutions for a Nonlocal Boundary-Value Problem with Vector-Valued Response

ABSTRACT

The existence of a countable set of positive solutions for a nonlocal boundary-value problem with vector-valued response is investigated by some variational methods based on the idea of the Fenchel conjugate. As a consequence of a duality developed here, we obtain the existence of a countable set of solutions for our problem that are minimizers to a certain integral functional. We derive (also in the superlinear case) a measure of a duality gap between primal and dual functional for approximate solutions.     

On the sufficiency of finite support duals in semi-infinite linear programming

We consider semi-infinite linear programs with countably many constraints indexed by the natural numbers. When the constraint space is the vector space of all real valued sequences, we show that the finite support (Haar) dual is equivalent to the algebraic Lagrangian dual of the linear program. This settles a question left open by Anderson and Nash (1987). This result implies that if there is a duality gap between the primal linear program and its finite support dual, then this duality gap cannot be closed by considering the larger space of dual variables that define the algebraic Lagrangian dual. However, if the constraint space corresponds to certain subspaces of all real-valued sequences, there may be a strictly positive duality gap with the finite support dual, but a zero duality gap with the algebraic Lagrangian dual.     

一类Lagrangian对偶问题的零对偶间隙性质及其最优路径的收敛性

针对一般的非线性规划问题,利用某些Lagrange型函数给出了一类Lagrangian对偶问题的一般模型,并证明它与原问题之间存在零对偶间隙．针对具体的一类增广La- grangian对偶问题以及几类由非线性卷积函数构成的Lagrangian对偶问题,详细讨论了零对偶间隙的存在性．进一步,讨论了在最优路径存在的前提下,最优路径的收敛性质．     

A note on duality gap in the simple plant location problem

This paper studies the duality gap in the simple plant location problem, and presents general formulas for the gap when certain complementary slackness conditions are satisfied. We show that the duality gap derived by Erlenkotter [A dual-based procedure for uncapacitated facility location, Operations Research 26 (1978) 992–1009], and which has been widely used in the literature, is a special case of the formulas presented here. A counterexample demonstrates that an underlying assumption in Erlenkotter may be violated. The results may be used to obtain improved lower bounds for branch-and-bound algorithms.     

Generating and measuring instances of hard semidefinite programs

Linear Programming, LP, problems with finite optimal value have a zero duality gap and a primal–dual strictly complementary optimal solution pair. On the other hand, there exist Semidefinite Programming, SDP, problems which have a nonzero duality gap (different primal and dual optimal values; not both infinite). The duality gap is assured to be zero if a constraint qualification, e.g., Slater’s condition (strict feasibility) holds. Measures of strict feasibility, also called distance to infeasibility, have been used in complexity analysis, and, it is known that (near) loss of strict feasibility results in numerical difficulties. In addition, there exist SDP problems which have a zero duality gap but no strict complementary primal–dual optimal solution. We refer to these problems as hard instances of SDP. The assumption of strict complementarity is essential for asymptotic superlinear and quadratic rate convergence proofs. In this paper, we introduce a procedure for generating hard instances of SDP with a specified complementarity nullity (the dimension of the common nullspace of primal–dual optimal pairs). We then show, empirically, that the complementarity nullity correlates well with the observed local convergence rate and the number of iterations required to obtain optimal solutions to a specified accuracy, i.e., we show, even when Slater’s condition holds, that the loss of strict complementarity results in numerical difficulties. We include two new measures of hardness that correlate well with the complementarity nullity.     

Imputing a variational inequality function or a convex objective function: A robust approach

To impute the function of a variational inequality and the objective of a convex optimization problem from observations of (nearly) optimal decisions, previous approaches constructed inverse programming methods based on solving a convex optimization problem [17], [7]. However, we show that, in addition to requiring complete observations, these approaches are not robust to measurement errors, while in many applications, the outputs of decision processes are noisy and only partially observable from, e.g., limitations in the sensing infrastructure. To deal with noisy and missing data, we formulate our inverse problem as the minimization of a weighted sum of two objectives: 1) a duality gap or Karush–Kuhn–Tucker (KKT) residual, and 2) a distance from the observations robust to measurement errors. In addition, we show that our method encompasses previous ones by generating a sequence of Pareto optimal points (with respect to the two objectives) converging to an optimal solution of previous formulations. To compare duality gaps and KKT residuals, we also derive new sub-optimality results defined by KKT residuals. Finally, an implementation framework is proposed with applications to delay function inference on the road network of Los Angeles, and consumer utility estimation in oligopolies.     

Fuzzy ordered distance measures

Xu and Chen (J Syst Sci Syst Eng 17:432–445, 2008) introduced a type of ordered weighted distance measures called ordered weighted distance (OWD) measures whose fundamental aspect is the reordering step, which can be used in many actual fields, including group decision making, medical diagnosis, data mining, and pattern recognition, etc. The OWD measures are very suitable to deal with situations where the input data are expressed in exact numerical values. In this paper, we consider situations with linguistic, interval or fuzzy preference information, and develop some fuzzy ordered distance measures, such as linguistic ordered weighted distance measure, uncertain ordered weighted distance measure, linguistic hybrid weighted distance measure, and uncertain hybrid weighted distance measure, etc. After that, based on hybrid weighted distance measures, we establish a consensus reaching process of group decision making with linguistic, interval, triangular or trapezoidal fuzzy preference information.     

Duality theory in fuzzy optimization problems formulated by the Wolfe’s primal and dual pair

The weak and strong duality theorems in fuzzy optimization problem based on the formulation of Wolfe’s primal and dual pair problems are derived in this paper. The solution concepts of primal and dual problems are inspired by the nondominated solution concept employed in multiobjective programming problems, since the ordering among the fuzzy numbers introduced in this paper is a partial ordering. In order to consider the differentiation of a fuzzy-valued function, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the Wolfe’s dual problem can be formulated by considering the gradients of differentiable fuzzy- valued functions. The concept of having no duality gap in weak and strong sense are also introduced, and the strong duality theorems in weak and strong sense are then derived naturally.     

A class of dual fuzzy dynamic programs

In this paper we propose a large class of fuzzy dynamic programs. By use of the notion of dual binary relation we define a dual fuzzy dynamic program in the class. We establish two duality theorems between primal and dual fuzzy dynamic programs. One is for the two-parametric recursive equations. The other is for the nonparametric. We specify maximum–minimum process and minimum–minimum process in fuzzy environment and multiplicative–multiplicative process in quasi-stochastic environment. It is shown that the duality theorems hold between primal and dual programs.     

Duality of graph invariants

We study a new set of duality relations between weighted,combinatoric invariants of a graph G.The dualities arise from a non-linear transform B,acting on the weight function p.We define B on a space of real-valued functions O and investigate its properties.We show that three invariants(the weighted independence number,the weighted Lovasz number,and the weighted fractional packing number) are fixed points of B~2,but the weighted Shannon capacity is not.We interpret these invariants in the study of quantum non-locality.