A chain wants to set up a single new facility in a planar market where similar facilities of competitors, and possibly of its own chain, are already present. Fixed demand points split their demand probabilistically over all facilities in the market proportionally with their attraction to each facility, determined by the different perceived qualities of … Read more
The problem of optimally designing a trajectory for a space mission is considered in this paper. Actual mission design is a complex, multi-disciplinary and multi-objective activity with relevant economic implications. In this paper we will consider some simplified models proposed by the European Space Agency as test problems for global optimization. We show that many … Read more
A common way to produce a convex relaxation of a Mixed Integer Quadratically Constrained Program (MIQCP) is to lift the problem into a higher dimensional space by introducing variables $Y_{ij}$ to represent each of the products $x_i x_j$ of variables appearing in a quadratic form. One advantage of such extended relaxations is that they can … Read more
The detection of gravitational waves is a long-awaited event in modern physics and, to achieve this challenging goal, detectors with high sensitivity are being used or are under development. In order to extract gravitational signals, emitted by coalescing binary systems of compact objects (neutron stars and/or black holes), from noisy data obtained by interferometric detectors, … Read more
PSwarm was developed originally for the global optimization of functions without derivatives and where the variables are within upper and lower bounds. The underlying algorithm used is a pattern search method, more specifically a coordinate search method, which guarantees convergence to stationary points from arbitrary starting points. In the (optional) search step of coordinate search, … Read more
This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of non-convexities: integer variables and non-convex quadratic constraints. To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the lift-and-project methodology. In particular, … Read more
Many industrial problems can be naturally formulated using Mixed Integer Nonlinear Programming (MINLP). Motivated by the demand for Open-Source solvers for real-world MINLP problems, we have developed a spatial Branch-and-Bound software package named COUENNE (Convex Over- and Under-ENvelopes for Nonlinear Estimation). In this paper, we present the structure of couenne and discuss in detail our … Read more
Non-Convex Quadratic Programming with Box Constraints is a fundamental NP-hard global optimisation problem. Recently, some authors have studied a certain family of convex sets associated with this problem. We prove several fundamental results concerned with these convex sets: we determine their dimension, characterise their extreme points and vertices, show their invariance under certain affine transformations, … Read more
Focusing on what an optimization problem may comply with, the so-called convergence conditions have been proposed and sequentially a stochastic optimization algorithm named as DSZ algorithm is presented in order to deal with both unconstrained and constrained optimizations. Its principle is discussed in the theoretical model of DSZ algorithm, from which we present a practical … Read more
Value-at-Risk (VaR) is an integral part of contemporary financial regulations. Therefore, the measurement of VaR and the design of VaR optimal portfolios are highly relevant problems for financial institutions. This paper treats a VaR constrained Markowitz style portfolio selection problem when the distribution of returns of the considered assets are given in the form of … Read more