Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Partially distributed outer approximation

This paper presents a novel partially distributed outer approximation algorithm, named PaDOA, for solving a class of structured mixed integer convex programming problems to global optimality. The proposed scheme uses an iterative outer approximation method for coupled mixed integer optimization problems with separable convex objective functions, affine coupling constraints, and compact domain. PaDOA proceeds by alternating between solving large-scale structured mixed-integer linear programming problems and partially decoupled mixed-integer nonlinear programming subproblems that comprise much fewer integer variables. We establish conditions under which PaDOA converges to global minimizers after a finite number of iterations and verify these properties with an application to thermostatically controlled loads and to mixed-integer regression

A scenario approach for non-convex control design

Randomized optimization is an established tool for control design with modulated robustness. While for uncertain convex programs there exist randomized approaches with efficient sampling, this is not the case for non-convex problems. Approaches based on statistical learning theory are applicable to non-convex problems, but they usually are conservative in terms of performance and require high sample complexity to achieve the desired probabilistic guarantees. In this paper, we derive a novel scenario approach for a wide class of random non-convex programs, with a sample complexity similar to that of uncertain convex programs and with probabilistic guarantees that hold not only for the optimal solution of the scenario program, but for all feasible solutions inside a set of a-priori chosen complexity. We also address measure-theoretic issues for uncertain convex and non-convex programs. Among the family of non-convex control- design problems that can be addressed via randomization, we apply our scenario approach to randomized Model Predictive Control for chance-constrained nonlinear control-affine systems.Comment: Submitted to IEEE Transactions on Automatic Contro

GALINI: an extensible solver for mixed-integer quadratically-constrained problems

Many industrial relevant optimization problems can be formulated as Mixed-Integer Quadratically Constrained Problems. This class of problems are difficult to solve because of 1) the non-convex bilinear terms 2) integer variables. This thesis develops the Python library \suspect{} for detecting special structure (monotonicity and convexity) of Pyomo models. This library can be extended to provide specialized detection for complex expressions. As a motivating example, we show how the library can be used to detect the convexity of the reciprocal of the log mean temperature difference. This thesis introduces GALINI: a novel solver that is easy to extend at runtime with new 1) cutting planes, 2) primal heuristics, 3) branching strategies, 4) node selection strategies, and 5) relaxations.GALINI uses Pyomo to represent optimization problems, this decision makes it possible to integrate with the existing Pyomo ecosystem to provide, for example, building blocks for relaxations. We test the solver on two large datasets and show that the performance is comparable to existing open source solvers. Finally, we present a library to formulate pooling problems, a class of network flow problems, using Pyomo. The library provides a mechanism to automatically generate the PQ-formulation for pooling problems. Since the library keeps the knowledge of the original network, it can 1) use a mixed-integer programming primal heuristic specialized for the pooling problem to find a feasible solution, and 2) generate valid cuts for the pooling problem. We use this library to develop an extension for GALINI that uses the mixed-integer programming primal heuristic to find a feasible solution and that generates cuts at every node of the branch & cut algorithm. We test GALINI with the pooling extensions on large scale instances of the pooling problem and show that we obtain results that are comparable to or better than the best available commercial solver on dense instances.Open Acces