Branch and Bound Based Coordinate Search Filter Algorithm for Nonsmooth Nonconvex Mixed-Integer Nonlinear Programming Problems

Abstract

Abstract. A mixed-integer nonlinear programming problem (MINLP) is a problem with continuous and integer variables and at least, one nonlinear function. This kind of problem appears in a wide range of real applications and is very difficult to solve. The difficulties are due to the nonlinearities of the functions in the problem and the integrality restrictions on some variables. When they are nonconvex then they are the most difficult to solve above all. We present a methodology to solve nonsmooth nonconvex MINLP problems based on a branch and bound paradigm and a stochastic strategy. To solve the relaxed subproblems at each node of the branch and bound tree search, an algorithm based on a multistart strategy with a coordinate search filter methodology is implemented. The produced numerical results show the robustness of the proposed methodology.

Interrupted searches in the BBMCSFilter context for MINLP problems

“…To analyze and compare the behavior of the new interrupted coordinate search strategy (within the BBMCSFilter algorithm) with the BBMCSFilter itself, we set all the parameters as presented in [4], and in equation (2) we set e = 0.1. The new algorithm is tested using a benchmark set of problems (from the engineering field) from the MINLPLib library available online [7].…”

“…A detail description of the BBMCSFilter algorithm is available in [4]. To settle the first issue pointed out above, the algorithm uses a BB paradigm.…”

“…Recently, the nonconvex MINLP research area became more interesting since metaheuristics and derivative-free algorithms, like the genetic algorithm, ant colony, differential evolution, pattern search algorithms, and the multistart Hooke and Jeeves algorithm [3] have been used in MINLP. (For details see the references included in [4]. )…”

“…In the BBMCSFilter method, the continuous NLP problems that appear in the BB tree search are solved to optimality by the MCSFilter method. This is a derivative-free global method that relies on a multistart algorithm coupled with a coordinate search filter method as the local procedure [4,5]. The MCSFilter has shown to be an efficient algorithm for finding multiple minima of nonsmooth and nonconvex NLP problems, and hence the global one [6].…”

“…In the present study we are particularly interested in solving this type of problems. In [4], the BBMCSFilter method, which relies on a BB framework and a derivative-free methodology to solve nonsmooth and nonconvex MINLP problems, is presented. This study aims to modify the BBMCSFilter method so that a reduction on the number of function evaluations and, consequently, on the CPU time, required to reach a global solution to the MINLP problems, is obtained.…”