An E-relaxation method for separable convex cost generalized network flow problems

Cover title. "The extended abstract of this article appeared in the proceedings of the 5th International IPCO Conference, Vancouver, June 1996."--P. 1.Includes bibliographical references (p. 18-21).Supported by the National Science Foundation. CCR-9311621, DMI-9300494by Paul Tseng, Dimitri P. Bertsekas

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

Concave Generalized Flows with Applications to Market Equilibria

We consider a nonlinear extension of the generalized network flow model, with the flow leaving an arc being an increasing concave function of the flow entering it, as proposed by Truemper and Shigeno. We give a polynomial time combinatorial algorithm for solving corresponding flow maximization problems, finding an epsilon-approximate solution in O(m(m+log n)log(MUm/epsilon)) arithmetic operations and value oracle queries, where M and U are upper bounds on simple parameters. This also gives a new algorithm for linear generalized flows, an efficient, purely scaling variant of the Fat-Path algorithm by Goldberg, Plotkin and Tardos, not using any cycle cancellations. We show that this general convex programming model serves as a common framework for several market equilibrium problems, including the linear Fisher market model and its various extensions. Our result immediately extends these market models to more general settings. We also obtain a combinatorial algorithm for nonsymmetric Arrow-Debreu Nash bargaining, settling an open question by Vazirani.Comment: Major revision. Instead of highest gain augmenting paths, we employ the Fat-Path framework. Many parts simplified, running time for the linear case improve

Fast exact algorithms for optimization problems in resource allocation and switched linear systems

University of Minnesota Ph.D. dissertation.June 2019. Major: Industrial Engineering. Advisor: Qie He. 1 computer file (PDF); x, 138 pages.Discrete optimization is a branch of mathematical optimization where some of the decision variables are restricted to real values in a discrete set. The use of discrete decision variables greatly expands the scope and capacity of mathematical optimization models. In the era of big data, efficiency and scalability are increasingly important in evaluating the performance of an algorithm. However, discrete optimization problems usually are challenging to solve. In this thesis, we develop new fast exact algorithms for discrete optimization problems arising in the field of resource allocation and switched linear systems. The first problem is the discrete resource allocation problem with nested bound constraints. It is a fundamental problem with a wide variety of applications in search theory, economics, inventory systems, etc. Given $B$ units of resource and $n$ activities, each of which associated with a convex allocation cost $f_i(\cdot)$, we aim to find an allocation of resources to the $n$ activities, denoted by \bm{x} \in \Ze^n, to minimize the total allocation cost $\sum\limits_{i = 1}^{n} f_i(x_i)$ subject to the total amount of resource constraint as well as lower and upper bound constraints on total resource allocated to subsets of activities. We develop a $\Theta(n^2\log\frac{B}{n})$-time algorithm for it. It is an infeasibility-guided divide-and-conquer algorithm and the worst-case complexity is usually not achieved. Numerical experiments demonstrate that our algorithm significantly outperforms a state-of-the-art optimization solver and the performance of our algorithm is competitive compared to the algorithm with the best worst-case complexity for this problem in the literature. The second problem is the minimum convex cost network flow problem on the dynamic lot size network. In the dynamic lot size network, there are one source node and $n$ sink nodes with demand $d_i, i = 1, \dots, n$. Let $B = \sum_{i=0}^{n}d_i$ be the total demand. We aim to find a flow $\bm{x}$ to minimize the total arc cost and satisfy all the flow balance and capacity constraints. Many optimization models in the literature can be seen as special cases of this problem, including dynamic lot-sizing problem and speed optimization. It is also a generalization of the first problem. We develop the Scaled Flow-improving Algorithm. For the continuous problem, our algorithm finds a solution that is at most $\epsilon$ away from an optimal solution in terms of the infinity norm in $O(n^2\log{\frac{B}{n\epsilon}})$ time. For the integer problem, our algorithm terminates in $O(n^2\log\frac{B}{n})$ time. Our algorithm has the best worst-case complexity in the literature. In particular, it solves the discrete resource allocation problem with nested bound constraints in $O(n\log{n}\log\frac{B}{n})$ time and it also achieves the best worst-case complexity for that problem. We conduct extensive numerical experiments on instances with a variety of convex objectives. The numerical result demonstrates the efficiency of our algorithm in solving large-sized instances. The last problem is the optimal control problem in switched linear systems. We consider the following dynamical system that consists of several linear subsystems: $K$ matrices, each chosen from the given set of matrices, to maximize a convex function over the product of the $K$ matrices and the given vector.This simple problem has many applications in operations research and control, yet a moderate-sized instance is challenging to solve to optimality for state-of-the-art optimization software. We prove the problem is NP-hard. We propose a simple exact algorithm for this problem. Our algorithm runs in polynomial time when the given set of matrices has the oligo-vertex property, a concept we introduce for a set of matrices. We derive several easy-to-verify sufficient conditions for a set of matrices to have the oligo-vertex property. In particular, we show that a pair of binary matrices has the oligo-vertex property. Numerical results demonstrate the clear advantage of our algorithm in solving large-sized instances of the problem over one state-of-the-art global solver. We also pose several open questions on the oligo-vertex property and discuss its potential connection with the finiteness property of a set of matrices, which may be of independent interest

Polynomial methods for separable convex optimization in unimodular linear spaces with applications to circulations and co-circulations in networks

Available at INIST (FR), Document Supply Service, under shelf-number : 17660, issue : a.1994 n.927-M / INIST-CNRS - Institut de l'Information Scientifique et TechniqueSIGLEFRFranc