Optimization under uncertainty and risk: Quadratic and copositive approaches

Robust optimization and stochastic optimization are the two main paradigms for dealing with the uncertainty inherent in almost all real-world optimization problems. The core principle of robust optimization is the introduction of parameterized families of constraints. Sometimes, these complicated semi-infinite constraints can be reduced to finitely many convex constraints, so that the resulting optimization problem can be solved using standard procedures. Hence flexibility of robust optimization is limited by certain convexity requirements on various objects. However, a recent strain of literature has sought to expand applicability of robust optimization by lifting variables to a properly chosen matrix space. Doing so allows to handle situations where convexity requirements are not met immediately, but rather intermediately. In the domain of (possibly nonconvex) quadratic optimization, the principles of copositive optimization act as a bridge leading to recovery of the desired convex structures. Copositive optimization has established itself as a powerful paradigm for tackling a wide range of quadratically constrained quadratic optimization problems, reformulating them into linear convex-conic optimization problems involving only linear constraints and objective, plus constraints forcing membership to some matrix cones, which can be thought of as generalizations of the positive-semidefinite matrix cone. These reformulations enable application of powerful optimization techniques, most notably convex duality, to problems which, in their original form, are highly nonconvex. In this text we want to offer readers an introduction and tutorial on these principles of copositive optimization, and to provide a review and outlook of the literature that applies these to optimization problems involving uncertainty

Robust topology optimization of three-dimensional photonic-crystal band-gap structures

We perform full 3D topology optimization (in which "every voxel" of the unit cell is a degree of freedom) of photonic-crystal structures in order to find optimal omnidirectional band gaps for various symmetry groups, including fcc (including diamond), bcc, and simple-cubic lattices. Even without imposing the constraints of any fabrication process, the resulting optimal gaps are only slightly larger than previous hand designs, suggesting that current photonic crystals are nearly optimal in this respect. However, optimization can discover new structures, e.g. a new fcc structure with the same symmetry but slightly larger gap than the well known inverse opal, which may offer new degrees of freedom to future fabrication technologies. Furthermore, our band-gap optimization is an illustration of a computational approach to 3D dispersion engineering which is applicable to many other problems in optics, based on a novel semidefinite-program formulation for nonconvex eigenvalue optimization combined with other techniques such as a simple approach to impose symmetry constraints. We also demonstrate a technique for \emph{robust} topology optimization, in which some uncertainty is included in each voxel and we optimize the worst-case gap, and we show that the resulting band gaps have increased robustness to systematic fabrication errors.Comment: 17 pages, 9 figures, submitted to Optics Expres

Nonconvex robust optimization

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2007.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 133-138).We propose a novel robust optimization technique, which is applicable to nonconvex and simulation-based problems. Robust optimization finds decisions with the best worst-case performance under uncertainty. If constraints are present, decisions should also be feasible under perturbations. In the real-world, many problems are nonconvex and involve computer-based simulations. In these applications, the relationship between decision and outcome is not defined through algebraic functions. Instead, that relationship is embedded within complex numerical models. Since current robust optimization methods are limited to explicitly given convex problems, they cannot be applied to many practical problems. Our proposed method, however, operates on arbitrary objective functions. Thus, it is generic and applicable to most real-world problems. It iteratively moves along descent directions for the robust problem, and terminates at a robust local minimum. Because the concepts of descent directions and local minima form the building blocks of powerful optimization techniques, our proposed framework shares the same potential, but for the richer, and more realistic, robust problem.(cont.) To admit additional considerations including parameter uncertainties and nonconvex constraints, we generalized the basic robust local search. In each case, only minor modifications are required - a testimony to the generic nature of the method, and its potential to be a component of future robust optimization techniques. We demonstrated the practicability of the robust local search technique in two realworld applications: nanophotonic design and Intensity Modulated Radiation Therapy (IMRT) for cancer treatment. In both cases, the numerical models are verified by actual experiments. The method significantly improved the robustness for both designs, showcasing the relevance of robust optimization to real-world problems.by Kwong Meng Teo.Ph.D

Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design

We describe an elementary algorithm to build convex inner approximations of nonconvex sets. Both input and output sets are basic semialgebraic sets given as lists of defining multivariate polynomials. Even though no optimality guarantees can be given (e.g. in terms of volume maximization for bounded sets), the algorithm is designed to preserve convex boundaries as much as possible, while removing regions with concave boundaries. In particular, the algorithm leaves invariant a given convex set. The algorithm is based on Gloptipoly 3, a public-domain Matlab package solving nonconvex polynomial optimization problems with the help of convex semidefinite programming (optimization over linear matrix inequalities, or LMIs). We illustrate how the algorithm can be used to design fixed-order controllers for linear systems, following a polynomial approach

Convex Relaxations and Approximations of Chance-Constrained AC-OPF Problems

This paper deals with the impact of linear approximations for the unknown nonconvex confidence region of chance-constrained AC optimal power flow problems. Such approximations are required for the formulation of tractable chance constraints. In this context, we introduce the first formulation of a chance-constrained second-order cone (SOC) OPF. The proposed formulation provides convergence guarantees due to its convexity, while it demonstrates high computational efficiency. Combined with an AC feasibility recovery, it is able to identify better solutions than chance-constrained nonconvex AC-OPF formulations. To the best of our knowledge, this paper is the first to perform a rigorous analysis of the AC feasibility recovery procedures for robust SOC-OPF problems. We identify the issues that arise from the linear approximations, and by using a reformulation of the quadratic chance constraints, we introduce new parameters able to reshape the approximation of the confidence region. We demonstrate our method on the IEEE 118-bus system