New Auction Algorithms for the Assignment Problem and Extensions

We consider the classical linear assignment problem, and we introduce new auction algorithms for its optimal and suboptimal solution. The algorithms are founded on duality theory, and are related to ideas of competitive bidding by persons for objects and the attendant market equilibrium, which underlie real-life auction processes. We distinguish between two fundamentally different types of bidding mechanisms: aggressive and cooperative. Mathematically, aggressive bidding relies on a notion of approximate coordinate descent in dual space, an epsilon-complementary slackness condition to regulate the amount of descent approximation, and the idea of epsilon-scaling to resolve efficiently the price wars that occur naturally as multiple bidders compete for a smaller number of valuable objects. Cooperative bidding avoids price wars through detection and cooperative resolution of any competitive impasse that involves a group of persons. We discuss the relations between the aggressive and the cooperative bidding approaches, we derive new algorithms and variations that combine ideas from both of them, and we also make connections with other primal-dual methods, including the Hungarian method. Furthermore, our discussion points the way to algorithmic extensions that apply more broadly to network optimization, including shortest path, max-flow, transportation, and minimum cost flow problems with both linear and convex cost functions

Towards auction algorithms for large dense assignment problems

Linear sum assignment problem, Auction algorithm, Linear optimization, Distributed algorithms,

Theoretical and computational advances in finite-size facility placement and assignment problems

The goal of this research is to develop fundamental theory and exact solution methods for the optimal placement of multiple, finite-size, rectangular facilities in presence of existing rectangular facilities, in a plane. Applications of this research can be found in facility layout (re)design in manufacturing, distribution systems, services (retail outlets, hospital floors, etc.), and printed circuit board design; where designing an efficient layout can save millions of dollars in operational costs. Main difficulty of this optimization problem lies in its continuous non-convex/non-concave feasible space, which makes it tough to escape local optimality. Through this research, novel approaches will be proposed which can be used to distill this continuous space into a finite set of candidate solutions, making it amenable to search for the global optimum. The first two parts of this research deal with establishing a unified theory for the finite-size facility placement problem and establishing the theory of dominance for pruning the sub-optimal solutions. Traditionally, the facility location/layout problems are modeled as the Quadratic Assignment Problem (QAP), which is strongly NP-hard. Also, for getting strong lower bounds in the dominance procedure, we may need to solve an instance of the NP-hard Quadratic Semi-Assignment Problem (QSAP). To this end, the third part of this research deals with investigating parallel and High Performance Computing (HPC) methods for solving the Linear Assignment Problem (LAP), which is an important sub-problem of the QAP. The final part of this research deals with investigating parallel and HPC methods for obtaining strong lower bounds and possibly solving large QAPs. Since QAP is known to be a computationally intensive problem, it should be noted that large in this context means problem instances with up to 30 facilities and locations