. Let F be a compact subset of the n-dimensional Euclidean space R n represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each of our methods generates a sequence of compact convex subsets C k (k = 1, 2, . . . ) of R n such that (a) the convex hull of F # C k+1 # C k (monotonicity), (b) # # k=1 C k = the convex hull of F (asymptotic convergence). Our methods are extensions of the corresponding Lovasz--Schrijver lift-and-project procedures with the use of SDP or LP relaxation applied to general quadratic optimization problems (QOPs) with infinitely many quadratic inequality constraints. Utilizing descriptions of sets based on cones of matrices and their duals, we establish the exact equivalence of the SDP relaxation and the semiinfinite convex QOP relaxation proposed originally by Fujie and Kojima. Using th...

Spurred by developments in spatial planning in robotics, computer graphics, and VLSI layout, considerable attention has been devoted recently to the problem of moving sets of objects, such as line segments and polygons in the plane to polyhedra in three dimensions, without allowing collisions between the objects. One class of such problems considers the separability of sets of objects under different kinds of motions and various definitions of separation. This paper surveys this new area of research in a tutorial fashion, present new results, and provides a list of open problems and suggestions for further research. Key Words and Phrases: sofa problem, polygons, polyhedra, movable separability, visibility hulls, hidden lines, hidden surfaces, algorithms, complexity, computational geometry, spatial planning, collision avoidance, robotics, artificial intelligence. CR Categories: 3.36, 3.63, 5.25. 5.32. 5.5 * Research supported by NSERC Grant no. A9293 and FCAR Grant no.EQ1678. - 2 - ...

The development of algorithms and software for the solution of large-scale optimization problems has been the main motivation behind the research on the identification properties of optimization algorithms. The aim of an identification result for a linearly constrained problem is to show that if the sequence generated by an optimization algorithm converges to a stationary point, then there is a nontrivial face F of the feasible set such that after a finite number of iterations, the iterates enter and remain in the face F . This paper develops the identification properties of linearly constrained optimization algorithms without any nondegeneracy or linear independence assumptions. The main result shows that the projected gradient converges to zero if and only if the iterates enter and remain in the face exposed by the negative gradient. This result generalizes results of Burke and Moré obtained for nondegenerate cases.

This paper deals with constrained optimization of Markov Decision Processes with a countable state space, compact action sets, continuous transition probabilities, and upper semi-continuous reward functions. The objective is to maximize the expected total discounted reward for one reward function, under several inequality constraints on similar criteria with other reward functions. Sippose a

With the aid of some novel complementarity constraint qualifications, we derive some simplied primal-dual characterizations of a B-stationary point for a mathematical program with complementarity constraints (MPEC). The approach is based on a locally equivalent piecewise formulation of such a program near a feasible point. The simplied results, which rely heavily on a careful dissection and improved understanding of the tangent cone of the feasible region of the program, bypass the combinatorial characterization that is intrinsic to B-stationarity.

Abstract. In this survey we review the many faces of the S-lemma, a result about the correctness of the S-procedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the S-lemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry. Key words. S-lemma, S-procedure, control theory, nonconvex theorem of alternatives, numerical range, relaxation theory, semidefinite optimization, generalized convexities

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. The paper describes and analyzes the cost approximation algorithm. This class of iterative descent algorithms for nonlinear programs and variational inequalities places a large number of algorithms within a common framework and provides a means for analyzing relationships among seemingly unrelated methods. A common property of the methods included in the framework is that their subproblems may be characterized by monotone mappings, which replace an additive part of the original cost mapping in an iterative manner; alternately, a step is taken in the direction obtained in order to reduce the value of a merit function for the original problem. The generality of the framework is illustrated through examples, and the convergence characteristics of the algorithm are analyzed for applications to nondifferentiable optimization. The convergence results are applied to some example methods, demonstrating the strength of the analysis compared to existing results. Key Words. Nondifferentiable o...

The coordination sequence fS(n)g of a lattice or net gives the number of nodes that are n bonds away from a given node. S(1) is the familiar coordination number. Extending work of O'Keeffe and others, we give explicit formulae for the coordination sequences of the root lattices A d , D d , E 6 , E 7 , E 8 and their duals. Proofs are given for many of the formulae, and for the fact that in every case S(n) is a polynomial in n, although some of the individual formulae are conjectural. In the majority of cases the set of nodes that are at most n bonds away from a given node form a polytopal cluster whose shape is the same as that of the contact polytope for the lattice. It is also shown that among all the Barlow packings in three dimensions the hexagonal close packing has the greatest coordination sequence, and the face-centered cubic lattice the smallest, as conjectured by O'Keeffe. 1. Introduction The coordination sequence of an infinite vertex-transitive graph G is the sequence fS...

Let X be a topological vector space, Y = IR n , n 2 IN, A a continuous linear map from X to Y , C ae X, B a convex set dense in C, and d 2 Y a data point. We derive conditions which guarantee that the set B " A \Gamma1 (d) is nonempty and dense in C " A \Gamma1 (d). Some applications to shape preserving interpolation and approximation are described. 1 Introduction A typical framework suitable for studying shape preserving interpolation and approximation can be described as follows. Let X be a Banach space and let A be a linear map from X to Y = IR n , n 2 IN. If d is a vector in Y , called a data point, then the unconstrained interpolation problem associated with the spaces X; Y , and the operator A, can be formulated as Find x 2 X such that Ax = d: (1) Usually, more than one solution exists, in which case one seeks a "best" solution based on predetermined criteria. For example, if k \Delta k is a (semi)norm on X , then an element x 0 2 X is sought such that kx 0 k = min...