## Invited Session Tue.3.H 1058

#### Tuesday, 15:15 - 16:45 h, Room: H 1058

**Cluster 10: Implementations & software** [...]

### NLP and MINLP software

**Chair: Hande Benson and Robert Vanderbei**

**Tuesday, 15:45 - 16:10 h, Room: H 1058, Talk 2**

**Klaus Schittkowski**

MISQP: A TR-SQP algorithm for the efficient solution of non-convex, non-relaxable mixed-integer nonlinear programming problems

**Coauthors: Oliver Exler, Thomas Lehmann**

**Abstract:**

We present a new sequential quadratic programming (SQP) algorithm stabilized by trust-regions for solving nonlinear, non-convex and non-relaxable mixed-integer optimization problems. The mixed-integer quadratic programming subproblems are solved by a branch-and-cut algorithm. Second order information is updated by a modified quasi-Newton update formula (BFGS) applied to the Lagrange function for continuous, but also for integer variables. The design goal is to solve practical optimization problems based on expensive executions of an underlying simulation program. Thus, the number of simulations or function evaluations, respectively, is our main performance criterion to measure the efficiency of the code. Numerical results are presented for a set of 175 mixed-integer test problems and different parameter settings of MISQP. The average total number of function evaluations of the new mixed-integer SQP code is about 1,200 including those needed for approximating partial derivatives.

**Tuesday, 16:15 - 16:40 h, Room: H 1058, Talk 3**

**Robert Vanderbei**

Fast fourier optimization

**Abstract:**

Many interesting and fundamentally practical optimization problems, ranging from optics, to signal processing, to radar and acoustics, involve constraints on the Fourier transform of a function. The fast Fourier transform (fft) is a well-known recursive algorithm that can dramatically improve the efficiency for computing the discrete Fourier transform. However, because it is recursive, it is difficult to embed into a linear optimization problem. In this talk, we explain the main idea behind the fast Fourier transform and show how to adapt it so as to make it encodable as constraints in an optimization problem. We demonstrate a real-world problem from the field of high-contrast imaging. On this problem, dramatic improvements are translated to an ability to solve problems with a much finer discretization. As we shall show, in general, the "fast Fourier'' version of the optimization constraints produces a larger but sparser constraint matrix and therefore one can think of the fast Fourier transform as a method of sparsifying the constraints in an optimization problem.