The linearization problem of a binary quadratic problem and its applications

We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be non-negative on the feasible set. Each linearization-based bound requires a set of linearizable matrices as an input. We prove that the Generalized Gilmore-Lawler bounding scheme for binary quadratic problems provides linearization-based bounds. Moreover, we show that the bound obtained from the first level reformulation linearization technique is also a type of linearization-based bound, which enables us to provide a comparison among mentioned bounds. However, the strongest linearization-based bound is the one that uses the full characterization of the set of linearizable matrices. Finally, we present a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives a complete characterization of the set of linearizable matrices for the quadratic shortest path problem

Robust Shortest Path Problem: Models and Solution Algorithms

Shortest path is a key component of several network related problems and has been widely used and applied in numerous disciplines such as transportation, logistics and telecommunication networks. This problem in the base deterministic settings lends itself to elegant and efficient solution methods. Nevertheless, the initial formulation is limited in number of ways; one such limitation is the need to accounting for inherent uncertainty in real world transportation networks. In recent years there has been a growing interest in incorporating uncertainty within the transportation network analysis models and particularly the shortest path problem. This dissertation contributes to the growing body of literature in dealing with uncertainty in shortest path problem by developing formulations as well as efficient solution methodologies for these class of problems.;There are number of approaches across the literature for incorporating the stochastic features of network related parameters such as travel time into the shortest path problem. One such approach is to minimize mean-risk analysis which is to minimize both the average cost and the risks arising from the uncertainty assumptions. The chief complication of such modeling approach is that the size of the nonlinear part of the objective function will increase for the large size real world network problem which undermines the efficiency of the existing solution approach. In response to such needs a solution methodology based on outer approximation (OA) strategy is proposed and customized which is highly efficient for real world large size instances.;In addition, in this dissertation a robust optimization approach for the shortest path problem where travel cost is uncertain and exact information on the distribution function is unavailable has been applied. Robust shortest path under such conditions is shown to be formulated as a binary nonlinear integer program, which can then be reformulated as a mixed integer conic quadratic program.;Finally, both two modeling frameworks provide a generalization in which links have two cost components, representing the expected cost and risk measure on the links -- the former term is additive, but the latter is not. In the third and final part of this dissertation, a solution methodology for general formulation of shortest path problem with non-additive continuous convex travel cost functions is presented

Computational Approaches for Stochastic Shortest Path on Succinct MDPs

We consider the stochastic shortest path (SSP) problem for succinct Markov decision processes (MDPs), where the MDP consists of a set of variables, and a set of nondeterministic rules that update the variables. First, we show that several examples from the AI literature can be modeled as succinct MDPs. Then we present computational approaches for upper and lower bounds for the SSP problem: (a)~for computing upper bounds, our method is polynomial-time in the implicit description of the MDP; (b)~for lower bounds, we present a polynomial-time (in the size of the implicit description) reduction to quadratic programming. Our approach is applicable even to infinite-state MDPs. Finally, we present experimental results to demonstrate the effectiveness of our approach on several classical examples from the AI literature

Strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives

A well-studied nonlinear extension of the minimum-cost flow problem is to minimize the objective $\sum_{ij\in E} C_{ij}(f_{ij})$ over feasible flows $f$, where on every arc $ij$ of the network, $C_{ij}$ is a convex function. We give a strongly polynomial algorithm for the case when all $C_{ij}$'s are convex quadratic functions, settling an open problem raised e.g. by Hochbaum [1994]. We also give strongly polynomial algorithms for computing market equilibria in Fisher markets with linear utilities and with spending constraint utilities, that can be formulated in this framework (see Shmyrev [2009], Devanur et al. [2011]). For the latter class this resolves an open question raised by Vazirani [2010]. The running time is $O(m^4\log m)$ for quadratic costs, $O(n^4+n^2(m+n\log n)\log n)$ for Fisher's markets with linear utilities and $O(mn^3 +m^2(m+n\log n)\log m)$ for spending constraint utilities. All these algorithms are presented in a common framework that addresses the general problem setting. Whereas it is impossible to give a strongly polynomial algorithm for the general problem even in an approximate sense (see Hochbaum [1994]), we show that assuming the existence of certain black-box oracles, one can give an algorithm using a strongly polynomial number of arithmetic operations and oracle calls only. The particular algorithms can be derived by implementing these oracles in the respective settings