An inexact strategy for the projected gradient algorithm in vector optimization problems on variable ordered spaces

Variable order structures model situations in which the comparison between two points depends on a point-to-cone map. In this paper, an inexact projected gradient method for solving smooth constrained vector optimization problems on variable ordered spaces is presented. It is shown that every accumulation point of the generated sequence satisfies the first order necessary optimality … Read more

On positive duality gaps in semidefinite programming

We study semidefinite programs (SDPs) with positive duality gaps, i.e., different optimal values in the primal and dual problems. the primal and dual problems differ. These SDPs are considered extremely pathological, they are often unsolvable, and they also serve as models of more general pathological convex programs. We first fully characterize two variable SDPs with … Read more

The Supporting Hyperplane Optimization Toolkit

In this paper, an open source solver for mixed-integer nonlinear programming (MINLP) problems is presented. The Supporting Hyperplane Optimization Toolkit (SHOT) combines a dual strategy based on polyhedral outer approximations (POA) with primal heuristics. The outer approximation is achieved by expressing the nonlinear feasible set of the MINLP problem with linearizations obtained with the extended … Read more

Tractable approximation of hard uncertain optimization problems

In many optimization problems uncertain parameters appear in a convex way, which is problematic as common techniques can only handle concave uncertainty. In this paper, we provide a systematic way to construct conservative approximations to such problems. Specifically, we reformulate the original problem as an adjustable robust optimization problem in which the nonlinearity of the … Read more

New sequential optimality conditions for mathematical problems with complementarity constraints and algorithmic consequences

In recent years, the theoretical convergence of iterative methods for solving nonlinear constrained optimization problems has been addressed using sequential optimality conditions, which are satisfied by minimizers independently of constraint qualifications (CQs). Even though there is a considerable literature devoted to sequential conditions for standard nonlinear optimization, the same is not true for Mathematical Problems … Read more

Quasi-Newton approaches to Interior Point Methods for quadratic problems

Interior Point Methods (IPM) rely on the Newton method for solving systems of nonlinear equations. Solving the linear systems which arise from this approach is the most computationally expensive task of an interior point iteration. If, due to problem’s inner structure, there are special techniques for efficiently solving linear systems, IPMs enjoy fast convergence and … Read more

On the Relation between MPECs and Optimization Problems in Abs-Normal Form

We show that the problem of unconstrained minimization of a function in abs-normal form is equivalent to identifying a certain stationary point of a counterpart Mathematical Program with Equilibrium Constraints (MPEC). Hence, concepts introduced for the abs-normal forms turn out to be closely related to established concepts in the theory of MPECs. We give a … Read more

An Enhanced Branch and Price Algorithm for the Time-Dependent Vehicle Routing Problem with Time Windows

In this paper we implement a branch and price (BP) algorithm for a time dependent vehicle routing problem with time windows (TDVRPTW) in which the goal is to minimize the total route duration (DM-TDVRPTW). The travel time between two customers depends on the departure time and, thus, it need not remain fixed along the planning … Read more

New facets for the consecutive ones polytope

A 0/1 matrix has the Consecutive Ones Property if a permutation of its columns exists such that the ones appear consecutively in each row. In many applications, one has to find a matrix having the consecutive ones property and optimizing a linear objective function. For this problem, literature proposes, among other approaches, an Integer Linear … Read more

A Lagrange decomposition based Branch and Bound algorithm for the Optimal Mapping of Cloud Virtual Machines

One of the challenges of cloud computing is to optimally and efficiently assign virtual machines to physical machines. The aim of telecommunication operators is to mini- mize the mapping cost while respecting constraints regarding location, assignment and capacity. In this paper we first propose an exact formulation leading to a 0-1 bilinear constrained problem. Then … Read more