We introduce a new efficient method to solve the continuous quadratic knapsack problem. This is a highly structured quadratic program that appears in different contexts. The method converges after O(n) iterations with overall arithmetic complexity O(n²). Numerical experiments show that in practice the method converges in a small number of iterations with overall linear complexity, … Read more
In this short and self contained paper we are proposing a simple penalty, which is a modification on the constraint set. We show that duality gap is eliminated under certain conditions. CitationMay,2010ArticleDownload View PDF
We consider the primal problem of finding the zeros of the sum of a maximally monotone operator with the composition of another maximally monotone operator with a linear continuous operator and a corresponding dual problem formulated by means of the inverse operators. A primal-dual splitting algorithm which simultaneously solves the two problems in finite-dimensional spaces … Read more
Farkas’ lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry: A linear matrix inequality … Read more
Mixed-integer conic programming is a generalization of mixed-integer linear programming. In this paper, we present an extension of the duality theory for mixed-integer linear programming to the case of mixed-integer conic programming. In particular, we construct a subadditive dual for mixed-integer conic programming problems. Under a simple condition on the primal problem, we are able … Read more
This paper studies dynamic stochastic optimization problems parametrized by a random variable. Such problems arise in many applications in operations research and mathematical finance. We give sufficient conditions for the existence of solutions and the absence of a duality gap. Our proof uses extended dynamic programming equations, whose validity is established under new relaxed conditions … Read more
Matrix rank minimization problems are gaining a plenty of recent attention in both mathematical and engineering fields. This class of problems, arising in various and across-discipline applications, is known to be NP-hard in general. In this paper, we aim at providing an approximation theory for the rank minimization problem, and prove that a rank minimization … Read more
In this paper new results are established in multiobjective DC programming with infinite convex constraints ($MOPIC$ for abbr.) that are defined on Banach space (finite or infinite) with objectives given as the difference of convex functions subject to infinite convex constraints. This problem can also be called multiobjective DC semi-infinite and infinite programming, where decision … Read more
The principle underlying this paper is the basic observation that the problem of simultaneously solving a large class of composite monotone inclusions and their duals can be reduced to that of finding a zero of the sum of a maximally monotone operator and a linear skew-adjoint operator. An algorithmic framework is developed for solving this … Read more
In this paper, with the help of convex-like function, we discuss the duality theory for nonconvex semidefinite programming. Our contributions are: duality theory for the general nonconvex semidefinite programming when Slater’s condition holds; perfect duality for a special case of the nonconvex semidefinite programming for which Slater’s condition fails. We point out that the results … Read more