Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

Optimization is at the core of control theory and appears in several areas of this field, such as optimal control, distributed control, system identification, robust control, state estimation, model predictive control and dynamic programming. The recent advances in various topics of modern optimization have also been revamping the area of machine learning. Motivated by the crucial role of optimization theory in the design, analysis, control and operation of real-world systems, this tutorial paper offers a detailed overview of some major advances in this area, namely conic optimization and its emerging applications. First, we discuss the importance of conic optimization in different areas. Then, we explain seminal results on the design of hierarchies of convex relaxations for a wide range of nonconvex problems. Finally, we study different numerical algorithms for large-scale conic optimization problems.Comment: 18 page

Exact Methods In Fractional Combinatorial Optimization

This dissertation considers a subclass of sum-of-ratios fractional combinatorial optimization problems (FCOPs) whose linear versions admit polynomial-time exact algorithms. This topic lies in the intersection of two scarcely researched areas of fractional programming (FP): sum-of-ratios FP and combinatorial FP. Although not extensively researched, the sum-of-ratios problems have a number of important practical applications in manufacturing, administration, transportation, data mining, etc. Since even in such a restricted research domain the problems are numerous, the main focus of this dissertation is a mathematical programming study of the three, probably, most classical FCOPs: Minimum Multiple Ratio Spanning Tree (MMRST), Minimum Multiple Ratio Path (MMRP) and Minimum Multiple Ratio Cycle (MMRC). The first two problems are studied in detail, while for the other one only the theoretical complexity issues are addressed. The dissertation emphasizes developing solution methodologies for the considered family of fractional programs. The main contributions include: (i) worst-case complexity results for the MMRP and MMRC problems; (ii) mixed 0-1 formulations for the MMRST and MMRC problems; (iii) a global optimization approach for the MMRST problem that extends an existing method for the special case of the sum of two ratios; (iv) new polynomially computable bounds on the optimal objective value of the considered class of FCOPs, as well as the feasible region reduction techniques based on these bounds; (v) an efficient heuristic approach; and, (vi) a generic global optimization approach for the considered class of FCOPs. Finally, extensive computational experiments are carried out to benchmark performance of the suggested solution techniques. The results confirm that the suggested global optimization algorithms generally outperform the conventional mixed 0{1 programming technique on larger problem instances. The developed heuristic approach shows the best run time, and delivers near-optimal solutions in most cases

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

Symmetry groups, semidefinite programs, and sums of squares

We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete symmetries in semidefinite programs. It is shown that symmetry exploitation allows a significant reduction in both matrix size and number of decision variables. This result is applied to semidefinite programs arising from the computation of sum of squares decompositions for multivariate polynomials. The results, reinterpreted from an invariant-theoretic viewpoint, provide a novel representation of a class of nonnegative symmetric polynomials. The main theorem states that an invariant sum of squares polynomial is a sum of inner products of pairs of matrices, whose entries are invariant polynomials. In these pairs, one of the matrices is computed based on the real irreducible representations of the group, and the other is a sum of squares matrix. The reduction techniques enable the numerical solution of large-scale instances, otherwise computationally infeasible to solve.Comment: 38 pages, submitte