McCormick-Based Relaxations of Algorithms

Abstract

Abstract. Theory and implementation for the global optimization of a wide class of algorithms is presented via convex/affine relaxations. The basis for the proposed relaxations is the systematic construction of subgradients for the convex relaxations of factorable functions by McCormick [Math. Prog., 10 (1976), pp. 147-175]. Similar to the convex relaxation, the subgradient propagation relies on the recursive application of a few rules, namely, the calculation of subgradients for addition, multiplication, and composition operations. Subgradients at interior points can be calculated for any factorable function for which a McCormick relaxation exists, provided that subgradients are known for the relaxations of the univariate intrinsic functions. For boundary points, additional assumptions are necessary. An automated implementation based on operator overloading is presented, and the calculation of bounds based on affine relaxation is demonstrated for illustrative examples. Two numerical examples for the global optimization of algorithms are presented. In both examples a parameter estimation problem with embedded differential equations is considered. The solution of the differential equations is approximated by algorithms with a fixed number of iterations. 1. Introduction. The development of deterministic algorithms based on continuous and/or discrete branch-and-bound [10,17,18] has facilitated the global optimization of nonconvex programs. The basic principle of branch-and-bound, and related algorithms such as branch-and-cut [19] and branch-and-reduce [27], is to bound the optimal objective value between a lower bound and an upper bound. By branching on the host set, these bounds become tighter and eventually converge. For minimization, upper bounds are typically obtained via a feasible point or via a local solution of the original program. For the lower bound, typically a convex or affine relaxation of the nonconvex program is constructed and solved to global optimality via a convex solver. Convex and concave envelopes or tight relaxations are known for a variety of simple nonlinear terms [1,33,35], and this allows the construction of convex and concave relaxations for a quite general class of functions through several methods [21,2,33,12]. Simple lower bounds from interval analysis are also widely used in global optimization, e.g., [6,7,25]. Such bounds are often weaker but less computationally expensive to evaluate than relaxation-based bounds. For instance, for a box-constrained problem, no linear program (LP) or convex nonlinear program (NLP) needs to be solved.The majority of the literature on global optimization considers nonconvex programs for which explicit functions are known for the objective and constraints. A more

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