A comparative study of SQP-type algorithms for nonlinear and nonconvex mixed-integer optimization

We present numerical results of a comparative study of codes for nonlinear and nonconvex mixed-integer optimization. The underlying algorithms are based on sequential quadratic programming (SQP) with stabilization by trust-regions, linear outer approximations, and branch-and-bound techniques. Themixed-integer quadratic programming subproblems are solved by a branch-and-cut algorithm. Second order information is updated by a quasi-Newton update formula (BFGS) applied to the Lagrange function for continuous, but also for integer variables. We do not require that the model functions can be evaluated at fractional values of the integer variables. Thus, partial derivatives with respect to integer variables are replaced by descent directions obtained from function values at neighboring grid points, and the number of simulations or function evaluations, respectively, is our main performance criterion to measure the efficiency of a code. Numerical results are presented for a set of 100 academic mixed-integer test problems. Since not all of our test examples are convex, we reach the best-known solutions in about 90 % of the test runs, but at least feasible solutions in the other cases. The average number of function evaluations of the new mixed-integer SQP code is between 240 and 500 including those needed for oneor two-sided approximations of partial derivatives or descent directions, respectively. In Sponsored by Shell GameChanger, SIEP Rijswijk, under project number 4600003917. O. Exler · K. Schittkowski (B) Department of Computer Science, University of Bayreuth, 95440 Bayreuth, Germany e-mail: [email protected] T. Lehmann Konrad-Zuse-Zentrum (ZIB), Takustr. 7, 14195 Berlin, Germany