Mixed-integer Convex Programming Research Articles

Cross-Layer Design for Lifetime Maximization in Interference-Limited Wireless Sensor Networks

We consider the joint optimal design of the physical, medium access control (MAC), and routing layers to maximize the lifetime of energy-constrained wireless sensor networks. The problem of computing lifetime-optimal routing flow, link schedule, and link transmission powers for all active time slots is formulated as a non-linear optimization problem. We first restrict the link schedules to the class of interference-free time division multiple access (TDMA) schedules. In this special case, we formulate the optimization problem as a mixed integerconvex program, which can be solved using standard techniques. Moreover, when the slots lengths are variable, the optimization problem is convex and can be solved efficiently and exactly using interior point methods. For general non-orthogonal link schedules, we propose an iterative algorithm that alternates between adaptive link scheduling and computation of optimal link rates and transmission powers for a fixed link schedule. The performance of this algorithm is compared to other design approaches for several network topologies. The results illustrate the advantages of load balancing, multihop routing, frequency reuse, and interference mitigation in increasing the lifetime of energy-constrained networks. We also briefly discuss computational approaches to extend this algorithm to large networks

Covariance-based hardware selection-Part I: globally optimal actuator selection

A new covariance based formulation of the actuator selection problem is presented. The proposed optimization problem is aimed at finding the set of minimum cost actuator arrays such that there exists a linear feedback for which all closed-loop signals will satisfy pre-specified variance bounds. Through a linear matrix inequality (LMI) based transformation we exactly convert the original problem into a computationally attractive mixed integer convex program (MICP). The resulting MICP leads to the first computational scheme capable of calculating globally optimal solutions to the covariance based actuator selection problem. It is further shown that the formulation is general enough to incorporate extensions to the actuator noise and actuator dynamics cases. Finally, a set of examples are presented to illustrate the scheme as well as the scalability of required computational effort.

Global Optimization of 0-1 Hyperbolic Programs

We develop eight different mixed-integer convex programming reformulations of 0-1 hyperbolic programs. We obtain analytical results on the relative tightness of these formulations and propose a branch and bound algorithm for 0-1 hyperbolic programs. The main feature of the algorithm is that it reformulates the problem at every node of the search tree. We demonstrate that this algorithm has a superior convergence behavior than directly solving the relaxation derived at the root node. The algorithm is used to solve a discrete i>p-choice facility location problem for locating ten restaurants in the city of Edmonton.

On the relationship between the integer and continuous solutions of convex programs

A bound is obtained in this note for the distance between the integer and real solutions to convex quadratic programs. This bound is a function of the condition number of the Hessian matrix. We further extend this proximity result to convex programs and mixed-integer convex programs. We also show that this bound is achievable in certain situations and the distance between the integer and continuous minimizers may tend to infinity.

Convex Methods in Sensor Placement

The goal of this work it is to reformulate a conceptually meaningful optimal sensor placement problem (as outlined by Bagajewicz, 1997). This reformulation converts the original mixed integer nonlinear program (MINLP) with dimensions dependent upon the value of the integer decision variables into a mixed integer convex program (MICP) with fixed matrix dimensions. This reformulation allows for the problem to be solved globally and efficiently using standard interior point and branch and bound search algorithms. Additionally, the traditional sensor placement problem, based on static process conditions, is extended to linear dynamic processes.

Ein algorithmus zur lösung gemisehtganzzahliger, konvexer optimierungsprobleme

For the solution of mixed integer convex programming problems with integer objective function an algorithm is established, which combines Kelley's cutting-plane provedure and Gomory's second algorithm. Under the assumptions to be made for these procedures a convergent algorithm results by introducing an additional cut. In this algorithm the number of constraints in the linear subproblems is bounded. In case of pure integer problems the algorithm becomes finite.