A wide variety of problems in control system theory fall within the class of parameterized Linear Matrix Inequalities (LMIs), that is, LMIs whose coefficients are functions of a parameter conned to a compact set. Such problems, though convex, involve an innite set of LMI constraints, hence are inherently difficult to solve numerically. This paper investigates relaxations of parameterized LMI problems into standard LMI problems using techniques relying on directional convexity concepts. An in-depth discussion of the impacts of the proposed techniques in quadratic programming, Lyapunov-based stability and performance analysis, µ analysis and Linear Parameter Varying control is provided. Illustrative examples are given to demonstrate the usefulness and practicality of the approach.

Many nonconvex nonlinear programming (NLP) problems of practical interest involve bilinear terms and linear constraints, as well as, potentially, other convex and nonconvex terms and constraints. In such cases, it may be possible to augment the formulation with additional linear constraints (a subset of Reformulation-Linearization Technique constraints) which do not a#ect the feasible region of the original NLP but tighten that of its convex relaxation to the extent that some bilinear terms may be dropped from the problem formulation. We present an e#cient graph-theoretical algorithm for e#ecting such exact reformulations of large, sparse NLPs. The global solution of the reformulated problem using spatial Branchand Bound algorithms is usually significantly faster than that of the original NLP. We illustrate this point by applying our algorithm to a set of pooling and blending global optimization problems.

. A wide variety of problems in control system theory fall within the class of parameterized Linear Matrix Inequalities (LMIs), that is, LMIs whose coefficients are functions of a parameter confined to a compact set. However, in contrast to LMIs, parameterized LMI (PLMIs) feasibility problems involve infinitely many LMIs hence are very hard to solve. In this paper, we propose several effective relaxation techniques to replace PLMIs by a finite set of LMIs. The resulting relaxed feasibility problems thus become convex and hence can be solved by very efficient interior point methods. Applications of these techniques to different problems such as robustness analysis, or Linear Parameter-Varying (LPV) control are then thoroughly discussed and illustrated by examples. 1 Introduction Linear matrix inequalities (LMIs) have emerged as a very powerful tool in the analysis and synthesis for robust control problems (see e.g. [6, 8, 12] and references therein). From a computational point of view,...

Many engineering optimization problems can be formulated as nonconvex nonlinear programming problems (NLPs) involving a nonlinear objective function subject to nonlinear constraints. Such problems may exhibit more than one locally optimal point. However, one is often solely or primarily interested in determining the globally optimal point. This thesis is concerned with techniques for establishing such global optima using spatial Branch-and-Bound (sBB) algorithms.

. This paper discusses computational complexity of conceptual successive convex relaxation methods proposed by Kojima and Tun¸cel for approximating a convex relaxation of a compact subset F = fx 2 C 0 : p(x) 0 (8p(\Delta) 2 P F )g of the n-dimensional Euclidean space R n . Here C 0 denotes a nonempty compact convex subset of R n , and P F a set of finitely or infinitely many quadratic functions. We evaluate the number of iterations which the successive convex relaxation methods require to attain a convex relaxation of F with a given accuracy ffl, in terms of ffl, the diameter of C 0 , the diameter of F , and some other quantities characterizing the Lipschitz continuity, the nonlinearity and the nonconvexity of the set P F of quadratic functions. Keywords: Complexity, Nonconvex Quadratic Program, Semidefinite Programming, Global Optimization, SDP Relaxation, Convex Relaxation, Lift-and-Project Procedure. 1 Introduction. In their paper [2], Kojima and Tun¸cel proposed a class of...

. The feasibility problem for constant scaling in output feedback control is considered. This is an inherently difficult problem [20, 21] since the set of feasible solutions is nonconvex and may be disconnected. Nevertheless, we show that this problem can be reduced to the global maximization of a concave function over a convex set, or alternatively, to the global minimization of a convex program with an additional reverse convex constraint. Thus this feasiblity problem belongs to the realm of d.c. optimization [14, 15, 32, 33], a new field which has recently emerged as an active promising research direction in nonconvex global optimization. By exploiting the specific d.c. structure of the problem, several algorithms are proposed which at every iteration require solving only either convex or linear subproblems. Analogous algorithms with new characterizations are proposed for the Bilinear Matrix Inequality (BMI) feasibility problem. 1 Introduction Consider the system given by Fig.1, ...

This article is concerned with two global optimization problems (P1) and (P2). Each of these problems is a fractional programming problem involving the maximization of a ratio of a convex function to a convex function, where at least one of the convex functions is a quadratic form. First, the article presents and validates a number of theoretical properties of these problems. Included among these properties is the result that, under a mild assumption, any globally optimal solution for problem (P1) must belong to the boundary of its feasible region. Also among these properties is a result that shows that problem (P2) can be reformulated as a convex maximization problem. Second, the article presents for the first time an algorithm for globally solving problem (P2). The algorithm is a branch and bound algorithm in which the main computational effort involves solving a sequence of convex programming problems. Convergence properties of the algorithm are presented, and computational issues that arise in implementing the algorithm are discussed. Preliminary indications are that the algorithm can be expected to provide a practical approach for solving problem (P2), provided that the number of variables is not too large.

We propose an algorithm to locate a global maximum of an increasing function subject to an increasing constraint on the cone of vectors with nonnegative coordinates. The algorithm is based on the outer approximation of the feasible set. We establish the convergence of the algorithm and provide a number of numerical experiments. We also discuss the types of constraints and objective functions for which the algorithm is best suited. Key words Monotonic global optimization, increasing functions, outer approximation method, abstract quasiconvexity. 1 Introduction The outer approximation method [1, 2, 10, 11, 12, 13] is a general approach for solving global optimization problems. Its successful implementation is based on specific properties of the problem under consideration. In particular the outer approximation method for concave minimization [1, 2, 12] exploits separation properties of convex sets and linear approximation properties of convex functions. It is well known that a lower se...

The problem of minimizing the ow value attained by maximal ows plays an important and interesting role to investigate how ine#ciently a network can be utilized. It is a typical multiextremal optimization problem, which can have local optima di#erent from global optima. We formulate this problem as an global optimization problem with a special structure and propose a method to combine di#erent techniques in local search and global optimization. Within the proposed algorithm, tha advantageous structure of network ow is fully exploited so that the algorithm should be suitable for handling the problem of moderate sizes. 1.

The paper discusses a general framework for outer approximation type algorithms for the canonical DC optimization problem. A thorough analysis of properties which guarantee convergence is carried out: different sets of general conditions are proposed and compared. They are exploited to build six different algorithms, which include the first cutting plane algorithm proposed by Tuy but also new ones. Approximate optimality conditions are introduced to guarantee the termination of the algorithms and the relationships with the global optimal value are discussed.