Rigorous Enclosures of Ellipsoids and Directed Cholesky Factorizations

Abstract

Abstract. This paper discusses the rigorous enclosure of an ellipsoid by a rectangular box, its interval hull, providing a convenient preprocessing step for constrained optimization problems.A quadratic inequality constraint with a positive definite Hessian defines an ellipsoid. The Cholesky factorization can be used to transform a strictly convex quadratic constraint into a norm inequality, for which the interval hull is easy to compute analytically. In exact arithmetic, the Cholesky factorization of a nonsingular symmetric matrix exists iff the matrix is positive definite. However, to cope efficiently with rounding errors in inexact arithmetic is nontrivial. Numerical tests show that even nearly singular problems can be handled successfully by our techniques.To rigorously account for the rounding errors involved in the computation of the interval hull and to handle quadratic inequality constraints having uncertain coefficients, we define the concept of a directed Cholesky factorization, and give two algorithms for computing one. We also discuss how a directed Cholesky factorization can be used for testing positive definiteness. Some numerical test are given in order to exploit the features and boundaries of the directed Cholesky factorization methods. , combine a number of methods and strategies to find one or more global solutions of a constrained optimization problem. Most of the techniques (e.g branch and bound, heuristics) require explicit bounds for each variable from below and from above. If a problem lacks these explicit prior bounds, the usual remedy is to set default upper and lower bounds on the variables, thereby changing the problem. If the global minimum lies outside the default bounds, the solver cannot find the solution.A rigorous enclosure technique for strictly convex quadratic constraints presented in this paper gives the possibility to obtain rigorous bounds on variables that are consequences of the constraints, without the need of giving explicit bounds on them. This makes the method a convenient preprocessing step for constrained optimization problems. On the other hand since the enclosures obtained by the method are rigorous, the method is also applicable in verified global optimization (e.g., Kearfott [10], Lebbah [11], Domes [7]) and in computer-assisted proofs (see, e.g., Neumaier [14]). Since it reduces the search space, it may also be important for stochastic, samplingbased optimization methods.The paper is logically divided into two parts. The first part (Sections 2 -4) is about computing rigorous enclosures for strictly convex quadratic constraints. In the second part (Sections 5 -7) the theory of the directed Cholesky factorization is developed as an essential tool for making the results of the first part rigorous.In the first section we find an optimal box enclosure of an ellipsoid defined by a simple Euclidean norm inequality constraint. In Section 2 we extend these results and

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“…There are several novel branching strategies within Couenne [27]. -GloptLAB [42,43,44,45] GloptLAB is a Matlab-based framework for solving quadratic constraint satisfaction problems [42]. The GloptLAB bounding and scaling strategies are particularly interesting [43,44,45].…”

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