An algorithm for constrained global optimization of multivariate polynomials using the Bernstein form and John optimality conditions

Abstract

We propose an algorithm for constrained global optimization of multivariate polynomials using the Bernstein form of polynomials. The proposed algorithm is of the branch and prune type, where branching is done using subdivision and pruning is done using the John optimality conditions for constrained minima. A main feature of this algorithm is that the branching and pruning operations are done with the Bernstein polynomial coeffi cients. The performance of the proposed algorithm is compared with those of existing global optimization techniques, on a few examples. The obtained results show the superiority of the proposed method over existing methods, in terms of number of iterations and computational time.

Constrained global optimization of multivariate polynomials using polynomial B-spline form and B-spline consistency prune approach

“…Compared with these methods the global optimization algorithm of multivariate polynomial using Bernstein form, e.g. [26,27] has the advantage that it avoids function evaluations which might be costly if the degree of the polynomial is high. Global optimization of polynomials using the Bernstein approach needs transformation of the given multivariate polynomial from its power form into its Bernstein form.…”

“…In this work, we propose B-spline based algorithms for solving non-convex nonlinear multivariate polynomial programming problems, where the objective function 𝑓 and constraints (𝑔 𝑖 & ℎ 𝑗 ) are limited to be polynomial functions. The proposed work extends the Bernstein method in [26,27] and B-spline method in [11][12][13] for constrained NLPs. The extensions are based on tools such as B-spline hull consistency (BsHC) and B-spline box consistency (BsBC) to contract the variable domains.…”