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

A novel dual-decomposition method for non-convex mixed integer quadratically constrained quadratic problems

In this paper, we propose the novel p-branch-and-bound method for solving two-stage stochastic programming problems whose deterministic equivalents are represented by non-convex mixed-integer quadratically constrained quadratic programming (MIQCQP) models. The precision of the solution generated by the p-branch-and-bound method can be arbitrarily adjusted by altering the value of the precision factor p. The proposed method combines two key techniques. The first one, named p-Lagrangian decomposition, generates a mixed-integer relaxation of a dual problem with a separable structure for a primal non-convex MIQCQP problem. The second one is a version of the classical dual decomposition approach that is applied to solve the Lagrangian dual problem and ensures that integrality and non-anticipativity conditions are met in the optimal solution. The p-branch-and-bound method's efficiency has been tested on randomly generated instances and demonstrated superior performance over commercial solver Gurobi. This paper also presents a comparative analysis of the p-branch-and-bound method efficiency considering two alternative solution methods for the dual problems as a subroutine. These are the proximal bundle method and Frank-Wolfe progressive hedging. The latter algorithm relies on the interpolation of linearisation steps similar to those taken in the Frank-Wolfe method as an inner loop in the classic progressive hedging.Comment: 19 pages, 5 table

Solution Techniques For Non-convex Optimization Problems

This thesis focuses on solution techniques for non-convex optimization problems. The first part of the dissertation presents a generalization of the completely positive reformulation of quadratically constrained quadratic programs (QCQPs) to polynomial optimization problems. We show that by explicitly handling the linear constraints in the formulation of the POP, one obtains a refinement of the condition introduced in Bai\u27s (2015) Thoerem on QCQPs, where the refined theorem only requires nonnegativity of polynomial constraints over the feasible set of the linear constraints. The second part of the thesis is concerned with globally solving non-convex quadratic programs (QPs) using integer programming techniques. More specifically, we reformulate non-convex QP as a mixed-integer linear problem (MILP) by incorporating the KKT condition of the QP to obtain a linear complementary problem, then use binary variables and big-M constraints to model the complementary constraints. We show how to impose bounds on the dual variables without eliminating all the (globally) optimal primal solutions; using some fundamental results on the solution of perturbed linear systems. The solution approach is implemented and labeled as quadprogIP, where computational results are presented in comparison with quadprogBB, BARON and CPLEX. The third part of the thesis involves the formulation and solution approach of a problem that arises from an on-demand aviation transportation network. A multi-commodity network flows (MCNF) model with side constraints is proposed to analyze and improve the efficiency of the on-demand aviation network, where the electric vertical-takeoff-and-landing (eVTOLs) transportation vehicles and passengers can be viewed as commodities, and routing them is equivalent to finding the optimal flow of each commodity through the network. The side constraints capture the decisions involved in the limited battery capacity for each eVTOL. We propose two heuristics that are efficient in generating integer feasible solutions that are feasible to the exponential number of battery side constraints. The last part of the thesis discusses a solution approach for copositive programs using linear semi-infinite optimization techniques. A copositive program can be reformulated as a linear semi-infinite program, which can be solved using the cutting plane approach, where each cutting plane is generated by solving a standard quadratic subproblem. Numerical results on QP-reformulated copositive programs are presented in comparison to the approximation hierarchy approach in Bundfuss (2009) and Yildirim (2012)