Continuous multifacility ordered median location problems

In this paper we propose a general methodology for solving a broad class of continuous, multifacility location problems, in any dimension and with ℓτ -norms proposing two different methodologies: 1) by a new second order cone mixed integer programming formulation and 2) by formulating a sequence of semidefinite programs that converges to the solution of the problem; each of these relaxed problems solvable with SDP solvers in polynomial time. We apply dimensionality reductions of the problems by sparsity and symmetry in order to be able to solve larger problems. Continuous multifacility location and Ordered median problems and Semidefinite programming and Moment problem.Junta de AndalucíaFondo Europeo de Desarrollo RegionalMinisterio de Ciencia e Innovació

A Semidefinite Programming approach for minimizing ordered weighted averages of rational functions

This paper considers the problem of minimizing the ordered weighted average (or ordered median) function of finitely many rational functions over compact semi-algebraic sets. Ordered weighted averages of rational functions are not, in general, neither rational functions nor the supremum of rational functions so that current results available for the minimization of rational functions cannot be applied to handle these problems. We prove that the problem can be transformed into a new problem embedded in a higher dimension space where it admits a convenient representation. This reformulation admits a hierarchy of SDP relaxations that approximates, up to any degree of accuracy, the optimal value of those problems. We apply this general framework to a broad family of continuous location problems showing that some difficult problems (convex and non-convex) that up to date could only be solved on the plane and with Euclidean distance, can be reasonably solved with different $\ell_p$-norms and in any finite dimension space. We illustrate this methodology with some extensive computational results on location problems in the plane and the 3-dimension space.Comment: 27 pages, 1 figure, 7 table

Conic Programming Approaches for Polynomial Optimization: Theory and Applications

Historically, polynomials are among the most popular class of functions used for empirical modeling in science and engineering. Polynomials are easy to evaluate, appear naturally in many physical (real-world) systems, and can be used to accurately approximate any smooth function. It is not surprising then, that the task of solving polynomial optimization problems; that is, problems where both the objective function and constraints are multivariate polynomials, is ubiquitous and of enormous interest in these fields. Clearly, polynomial op- timization problems encompass a very general class of non-convex optimization problems, including key combinatorial optimization problems.The focus of the first three chapters of this document is to address the solution of polynomial optimization problems in theory and in practice, using a conic optimization approach. Convex optimization has been well studied to solve quadratic constrained quadratic problems. In the first part, convex relaxations for general polynomial optimization problems are discussed. Instead of using the matrix space to study quadratic programs, we study the convex relaxations for POPs through a lifted tensor space, more specifically, using the completely positive tensor cone and the completely positive semidefinite tensor cone. We show that tensor relaxations theoretically yield no-worse global bounds for a class of polynomial optimization problems than relaxation for a QCQP reformulation of the POPs. We also propose an approximation strategy for tensor cones and show empirically the advantage of the tensor relaxation.In the second part, we propose an alternative SDP and SOCP hierarchy to obtain global bounds for general polynomial optimization problems. Comparing with other existing SDP and SOCP hierarchies that uses higher degree sum of square (SOS) polynomials and scaled diagonally sum of square polynomials (SDSOS) when the hierarchy level increases, these proposed hierarchies, using fixed degree SOS and SDSOS polynomials but more of these polynomials, perform numerically better. Numerical results show that the hierarchies we proposed have better performance in terms of tightness of the bound and solution time compared with other hierarchies in the literature.The third chapter deals with Alternating Current Optimal Power Flow problem via a polynomial optimization approach. The Alternating Current Optimal Power Flow (ACOPF) problem is a challenging non-convex optimization problem in power systems. Prior research mainly focuses on using SDP relaxations and SDP-based hierarchies to address the solution of ACOPF problem. In this Chapter, we apply existing SOCP hierarchies to this problem and explore the structure of the network to propose simplified hierarchies for ACOPF problems. Compared with SDP approaches, SOCP approaches are easier to solve and can be used to approximate large scale ACOPF problems.The last chapter also relates to the use of conic optimization techniques, but in this case to pricing in markets with non-convexities. Indeed, it is an application of conic optimization approach to solve a pricing problem in energy systems. Prior research in energy market pricing mainly focus on linear costs in the objective function. Due to the penetration of renewable energies into the current electricity grid, it is important to consider quadratic costs in the objective function, which reflects the ramping costs for traditional generators. This study address the issue how to find the market clearing prices when considering quadratic costs in the objective function

Solution Techniques For Non-convex Optimization Problems

This thesis focuses on solution techniques for non-convex optimization problems. The first part of the dissertation presents a generalization of the completely positive reformulation of quadratically constrained quadratic programs (QCQPs) to polynomial optimization problems. We show that by explicitly handling the linear constraints in the formulation of the POP, one obtains a refinement of the condition introduced in Bai\u27s (2015) Thoerem on QCQPs, where the refined theorem only requires nonnegativity of polynomial constraints over the feasible set of the linear constraints. The second part of the thesis is concerned with globally solving non-convex quadratic programs (QPs) using integer programming techniques. More specifically, we reformulate non-convex QP as a mixed-integer linear problem (MILP) by incorporating the KKT condition of the QP to obtain a linear complementary problem, then use binary variables and big-M constraints to model the complementary constraints. We show how to impose bounds on the dual variables without eliminating all the (globally) optimal primal solutions; using some fundamental results on the solution of perturbed linear systems. The solution approach is implemented and labeled as quadprogIP, where computational results are presented in comparison with quadprogBB, BARON and CPLEX. The third part of the thesis involves the formulation and solution approach of a problem that arises from an on-demand aviation transportation network. A multi-commodity network flows (MCNF) model with side constraints is proposed to analyze and improve the efficiency of the on-demand aviation network, where the electric vertical-takeoff-and-landing (eVTOLs) transportation vehicles and passengers can be viewed as commodities, and routing them is equivalent to finding the optimal flow of each commodity through the network. The side constraints capture the decisions involved in the limited battery capacity for each eVTOL. We propose two heuristics that are efficient in generating integer feasible solutions that are feasible to the exponential number of battery side constraints. The last part of the thesis discusses a solution approach for copositive programs using linear semi-infinite optimization techniques. A copositive program can be reformulated as a linear semi-infinite program, which can be solved using the cutting plane approach, where each cutting plane is generated by solving a standard quadratic subproblem. Numerical results on QP-reformulated copositive programs are presented in comparison to the approximation hierarchy approach in Bundfuss (2009) and Yildirim (2012)