Timo Berthold

Parallelization of the FICO Xpress-Optimizer

Computing hardware has mostly thrashed out the physical limits for speeding up individual computing cores. Consequently, the main line of progress for new hardware is growing the number of computing cores within a single CPU. This makes the study of efficient parallelization schemes for computation-intensive algorithms more and more important. A natural precondition to achieving reasonable speedups from parallelization is maintaining a high workload of the available computational resources. At the same time, reproducibility and reliability are key requirements for software that is used in industrial applications. In this paper, we present the new parallelization concept for the state-of-the-art MIP solver FICO Xpress-Optimizer. MIP solvers like Xpress are expected to be deterministic. This inevitably results in synchronization latencies which render the goal of a satisfying workload a challenge in itself. We address this challenge by following a partial information approach and separating the concepts of simultaneous tasks and independent threads from each other. Our computational results indicate that this leads to a much higher CPU workload and thereby to an improved, almost linear, scaling on modern high-performance CPUs. As an added value, the solution path that Xpress takes is not only deterministic in a fixed environment, but also, to a certain extent, thread-independent. This paper is an extended version of Berthold et al. [Parallelization of the FICO Xpress-Optimizer, in Mathematical Software – ICMS 2016: 5th International Conference, G.-M. Greuel, T. Koch, P. Paule, and A. Sommere, eds., Springer International Publishing, Berlin, 2016, pp. 251–258] containing more detailed technical descriptions, illustrative examples and updated computational results.

Three ideas for a feasibility pump for nonconvex MINLP

We describe an implementation of the Feasibility Pump heuristic for nonconvex MINLPs. Our implementation takes advantage of three novel techniques, which we discuss here: a hierarchy of procedures for obtaining an integer solution, a generalized definition of the distance function that takes into account the nonlinear character of the problem, and the insertion of linearization cuts for nonconvex constraints at every iteration. We implemented this new variant of the Feasibility Pump as part of the global optimization solver Couenne. We present experimental results that compare the impact of the three discussed features on the ability of the Feasibility Pump to find feasible solutions and on the solution quality.

Algorithms for discrete nonlinear optimization in FICO Xpress

The FICO Xpress-Optimizer is a commercial optimization solver for linear programming (LP), mixed integer linear programming (MIP), convex quadratic programming (QP), convex quadratically constrained quadratic programming (QCQP), second-order cone programming (SOCP) and their mixed-integer counterparts. Xpress also includes a general purpose non-linear solver, Xpress-NonLinear, which features a successive linear programming algorithm (SLP, first-order method), interior point methods and Artelys Knitro (second-order methods). This work explores algorithms for mixed-integer nonlinear programming problems (MINLPs), which are NP-hard in general, then it presents applications in signal processing and capitalizes advances in solving these problems with Xpress and its comprehensive suite of high-performance nonlinear solvers. Computational results show that signal processing nonlinear problems can be solved quickly and accurately, taking advantage of the algebraic modeling and procedural programming language, Xpress-Mosel, that allows to interact with the Xpress solver engines in a easy-to-learn way, and its unified modeling interface for all solvers, from linear to general nonlinear solvers.

Four Good Reasons to Use an Interior Point Solver Within a MIP Solver

“Interior point algorithms are a good choice for solving pure LPs or QPs, but when you solve MIPs, all you need is a dual simplex” This is the common conception which disregards that an interior point solution provides some unique structural insight into the problem at hand. In this paper, we will discuss some of the benefits that an interior point solver brings to the solution of difficult MIPs within FICO Xpress. This includes many different components of the MIP solver such as branching variable selection, primal heuristics, preprocessing, and of course the solution of the LP relaxation.