Nonmonotone line search methods with variable sample size

Abstract

© 2014, Springer Science+Business Media New York. Nonmonotone line search methods for unconstrained minimization with the objective functions in the form of mathematical expectation are considered. The objective function is approximated by the sample average approximation (SAA) with a large sample of fixed size. The nonmonotone line search framework is embedded with a variable sample size strategy such that different sample size at each iteration allow us to reduce the cost of the sample average approximation. The variable sample scheme we consider takes into account the decrease in the approximate objective function and the quality of the approximation of the objective function at each iteration and thus the sample size may increase or decrease at each iteration. Nonmonotonicity of the line search combines well with the variable sample size scheme as it allows more freedom in choosing the search direction and the step size while the sample size is not the maximal one and increases the chances of finding a global solution. Eventually the maximal sample size is used so the variable sample size strategy generates the solution of the same quality as the SAA method but with significantly smaller number of function evaluations. Various nonmonotone strategies are compared on a set of test problems.