Approximating Incremental Combinatorial Optimization Problems

We consider incremental combinatorial optimization problems, in which a solution is constructed incrementally over time, and the goal is to optimize not the value of the final solution but the average value over all timesteps. We consider a natural algorithm of moving towards a global optimum solution as quickly as possible. We show that this algorithm provides an approximation guarantee of (9+sqrt(21))/15 > 0.9 for a large class of incremental combinatorial optimization problems defined axiomatically, which includes (bipartite and non-bipartite) matchings, matroid intersections, and stable sets in claw-free graphs. Furthermore, our analysis is tight

Approximating incremental combinatorial optimization problems

We consider incremental combinatorial optimization problems, in which a solution is constructed incrementally over time, and the goal is to optimize not the value of the final solution but the average value over all timesteps. We consider a natural algorithm of moving towards a global optimum solution as quickly as possible. We show that this algorithm provides an approximation guarantee of (9 + √21)/15 > 0.9 for a large class of incremental combinatorial optimization problems defined axiomatically, which includes (bipartite and non-bipartite) matchings, matroid intersections, and stable sets in claw-free graphs. Furthermore, our analysis is tight

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 Polyhedral Approximation Framework for Convex and Robust Distributed Optimization

In this paper we consider a general problem set-up for a wide class of convex and robust distributed optimization problems in peer-to-peer networks. In this set-up convex constraint sets are distributed to the network processors who have to compute the optimizer of a linear cost function subject to the constraints. We propose a novel fully distributed algorithm, named cutting-plane consensus, to solve the problem, based on an outer polyhedral approximation of the constraint sets. Processors running the algorithm compute and exchange linear approximations of their locally feasible sets. Independently of the number of processors in the network, each processor stores only a small number of linear constraints, making the algorithm scalable to large networks. The cutting-plane consensus algorithm is presented and analyzed for the general framework. Specifically, we prove that all processors running the algorithm agree on an optimizer of the global problem, and that the algorithm is tolerant to node and link failures as long as network connectivity is preserved. Then, the cutting plane consensus algorithm is specified to three different classes of distributed optimization problems, namely (i) inequality constrained problems, (ii) robust optimization problems, and (iii) almost separable optimization problems with separable objective functions and coupling constraints. For each one of these problem classes we solve a concrete problem that can be expressed in that framework and present computational results. That is, we show how to solve: position estimation in wireless sensor networks, a distributed robust linear program and, a distributed microgrid control problem.Comment: submitted to IEEE Transactions on Automatic Contro

Global Optimization of Monotonic Programs: Applications in Polynomial and Stochastic Programming.

Monotonic optimization consists of minimizing or maximizing a monotonic objective function over a set of constraints defined by monotonic functions. Many optimization problems in economics and engineering often have monotonicity while lacking other useful properties, such as convexity. This thesis is concerned with the development and application of global optimization algorithms for monotonic optimization problems. First, we propose enhancements to an existing outer-approximation algorithm | called the Polyblock Algorithm | for monotonic optimization problems. The enhancements are shown to significantly improve the computational performance of the algorithm while retaining the convergence properties. Next, we develop a generic branch-and-bound algorithm for monotonic optimization problems. A computational study is carried out for comparing the performance of the Polyblock Algorithm and variants of the proposed branch-and-bound scheme on a family of separable polynomial programming problems. Finally, we study an important class of monotonic optimization problems | probabilistically constrained linear programs. We develop a branch-and-bound algorithm that searches for a global solution to the problem. The basic algorithm is enhanced by domain reduction and cutting plane strategies to reduce the size of the partitions and hence tighten bounds. The proposed branch-reduce-cut algorithm exploits the monotonicity properties inherent in the problem, and requires the solution of only linear programming subproblems. We provide convergence proofs for the algorithm. Some illustrative numerical results involving problems with discrete distributions are presented.Ph.D.Committee Chair: Al-Khayyal, Faiz; Committee Co-Chair: Ahmed, Shabbir; Committee Member: Barnes, Earl; Committee Member: Realff, Matthew; Committee Member: Shapiro, Ale