The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

http://egon.cheme.cmu.edu/Papers/BatchReviewMendezGrossmann.pdf

State-of-the-art review of optimization methods for short-term scheduling of batch processes

A survey for the quadratic assignment problem

The quadratic assignment problem (QAP) was introduced by Koopmans and Beckmann in 1957 as a mathematical model for the location of a set of indivisible economical activities [113]. Consider the problem of allocating a set of facilities to a set of locations, with the cost being a function of the distance and flow between the facilities, plus costs associated with a facility being placed at a certain location. The objective is to assign each facility to a location such that the total cost is minimized. Specifically, we are given three n x n input matrices with real elements F = (f ij ), D = (d kl ) and B = (b ik ), where f ij is the flow between the facility i and facility j, d kl is the distance between the location k and location l, and b ik is the cost of placing facility i at location k. The Koopmans-Beckmann version of the QAP can be formulated as follows: Let n be the number of facilities and locations and denote by N the set N = {1, 2,..., n}.

Quadratic Assignment Problem.

The max-sum labeling problem, defined as maximizing a sum of binary (ie, pairwise) functions of discrete variables, is a general NP-hard optimization problem with many applications, such as computing the MAP configuration of a Markov random field We review a not widely known approach to the problem, developed by Ukrainian researchers Schlesinger et al in 1976, and show how it contributes to recent results, most importantly, those on the convex combination of trees and tree-reweighted max-product In particular, we review Schlesinger et al's upper bound on the max-sum criterion, its minimization by equivalent transformations, its relation to the constraint satisfaction problem, the fact that this minimization is dual to a linear programming relaxation of the original problem, and the three kinds of consistency necessary for optimality of the upper bound We revisit problems with Boolean variables and supermodular problems We describe two algorithms for decreasing the upper bound We present an example application for structural image analysis

/pdf/a-linear-programming-approach-to-max-sum-problem-a-review-1i2n5dqatc.pdf

A Linear Programming Approach to Max-Sum Problem: A Review

Perspectives of Monge properties in optimization

Consider a network
 
$$\mathcal{N}$$

 =(G, c, τ) whereG=(N, A) is a directed graph andc

 ij
 andτ

 ij
 , respectively, denote the capacity and the transmission time of arc (i, j) ∈A. The quickest flow problem is then to determine for a given valueυ the minimum numberT(υ) of time units that are necessary to transmit (send)υ units of flow in
 
$$\mathcal{N}$$

 from a given sources to a given sinks′. In this paper we show that the quickest flow problem is closely related to the maximum dynamic flow problem and to linear fractional programming problems. Based on these relationships we develop several polynomial algorithms and a strongly polynomial algorithm for the quickest flow problem. Finally we report computational results on the practical behaviour of our metholds. It turns out that some of them are practically very efficient and well-suited for solving large problem instances.

The Quickest Flow Problem

A dynamic network consists of a directed graph with capacities, costs, and integral transit times on the arcs. In the minimum-cost dynamic flow problem (MCDFP), the goal is to compute, for a given dynamic network with source s, sink t, and two integers v and T, a feasible dynamic flow from s to t of value v, obeying the time bound T, and having minimum total cost. MCDFP contains as subproblems the minimum-cost maximum dynamic flow problem, where v is fixed to the maximum amount of flow that can be sent from s to t within time T and the minimum-cost quickest flow problem, where is T is fixed to the minimum time needed for sending v units of flow from s to t. We first prove that both subproblems are NP-hard even on two-terminal series-parallel graphs with unit capacities. As main result, we formulate a greedy algorithm for MCDFP and provide a full characterization via forbidden subgraphs of the class of graphs, for which this greedy algorithm always yields an optimum solution (for arbitrary choices of problem parameters). G is a subclass of the class of two-terminal series-parallel graphs. We show that the greedy algorithm solves MCDFP restricted to graphs in G in polynomial time.

Minimum-cost dynamic flows: The series-parallel case†

Permuting matrices to avoid forbidden submatrices

A dynamic network consists of a directed graph with capacities, costs and integral transit times on the arcs. In the minimum cost dynamic flow problem MCDFP, the goal is to compute for a given dynamic network with source s, sink t and two integers υ and T, a feasible dynamic flow from s to t of value υ, obeying the time bound T, and having minimum total cost. MCDFP contains as subproblems the minimum cost maximum dynamic flow problem (fix υ to the maximum amount of flow that can be sent from s to t within time T), and the minimum cost quickest flow problem (fix T to the minimum time needed for sending υ units of flow from s to t). We first prove that both subproblems are NP-hard even on two-terminal series-parallel graphs with unit capacities. As main result, we formulate a greedy algorithm for MCDFP and provide a full characterization via forbidden subgraphs of the class G of graphs, for which this greedy algorithm always yields an optimum solution (for arbitrary choices of problem parameters). G is a subclass of the class of two-terminal series-parallel graphs. It is shown that the greedy algorithm solves MCDFP restricted to graphs in G in polynomial time.

Bettina Klinz

Papers

Perspectives of Monge properties in optimization

The Quickest Flow Problem

Minimum-cost dynamic flows: The series-parallel case†

Permuting matrices to avoid forbidden submatrices

Minimum Cost Dynamic Flows: The Series-Parallel Case