Improving Hit-and-Run for global optimization

Abstract

Improving Hit-and-Run is a random search algorithm for global optimization that at each iteration generates a candidate point for improvement that is uniformly distributed along a randomly chosen direction within the feasible region. The candidate point is accepted as the next iterate if it offers an improvement over the current iterate. We show that for positive definite quadratic programs, the expected number of function evaluations needed to arbitrarily well approximate the optimal solution is at most O(n 5~2) where n is the dimension of the problem. Improving Hit-and-Run when applied to global optimization problems can therefore be expected to converge polynomially fast as it approaches the global optimum.

A generalization of the Solis–Wets method

“…The generality of the master method is considerable including, for instance, the conceptual algorithm of Solis and Wets (1981) and the improving hit-and-run algorithm (Zabinsky et al, 1993). Another specific embodiment of the master method is provided by the stochastic zigzag method-an optimization algorithm which is based on the works of Mexia et al (1999) and Pereira and Mexia (2010).…”

“…Other variants of the stochastic zigzag algorithm are also included in the general method; for example, if c¼2, we get the improving hit-and-run algorithm (Zabinsky et al, 1993;Andersen and Diaconis, 2007). We can also consider an alternative shape for the line, and the generality of the master method is such that we may even take a different shape per course.…”

Towards Pure Adaptive Search

“…These are the Improving Hit-and-Run algorithm (14], and the Hide-and-Seek algorithm [3]. The purpose of this paper is to approach .the problem from a third perspective, which we now describe.…”

Simulated annealing for complex portfolio selection problems

“…Lov ¶ asz and Simonovits [19] or Zabinsky et al [29]). The euclidean length of each move is now simply determined by the radius of the ball.…”

“…Due to the success of simulated annealing in this framework, several researchers have attempted to extend the approach to continuous minimization problems (see van Laarhoven and Aarts [27], Dekkers and Aarts [9], CSEP [6], Zabinsky et al [29]). However, few practical applications appear in the literature.…”