In this chapter we deal with the following nonlinear fractional programming problem: 

$$P:\mathop{{\max }}\limits_{{x \in s}} q(x) = (f(x) + \alpha )/((x) + \beta )$$

where f, g: R n → R, α, β ∈ R, S ⊆ R n . To simplify things, and without restricting the generality of the problem, it is usually assumed that, g(x) + β + 0 for all x ∈ S,S is non-empty and that the objective function has a finite optimal value.

Nonlinear Fractional Programming

Introduction. 1. Mathematical Preliminaries. Submodular Systems and Base Polyhedra. 2. From Matroids to Submodular Systems. Matroids. Polymatroids. Submodular Systems. 3. Submodular Systems and Base Polyhedra. Fundamental Operations on Submodular Systems. Greedy Algorithm. Structures of Base Polyhedra. Intersecting- and Crossing-Submodular Functions. Related Polyhedra. Submodular Systems of Network Type. Neoflows. 4. The Intersection Problem. The Intersection Theorem. The Discrete Separation Theorem. The Common Base Problem. 5. Neoflows. The Equivalence of the Neoflow Problems. Feasibility for Submodular Flows. Optimality for Submodular Flows. Algorithms for Neoflows. Matroid Optimization. Submodular Analysis. 6. Submodular Functions and Convexity. Conjugate Functions and a Fenchel-Type Min-Max Theorem for Submodular and Supermodular Functions. Subgradients of Submodular Functions. The Lovasz Extensions of Submodular Functions. 7. Submodular Programs. Submodular Programs - Unconstrained Optimization. Submodular Programs - Constrained Optimization. Nonlinear Optimization with Submodular Constraints. 8. Separable Convex Optimization. Optimality Conditions. A Decomposition Algorithm. Discrete Optimization. 9. The Lexicographically Optimal Base Problem. Nonlinear Weight Functions. Linear Weight Functions. 10. The Weighted Max-Min and Min-Max Problems. Continuous Variables. Discrete Variables. 11. The Fair Resource Allocation Problem. Continuous Variables. Discrete Variables. 12. The Neoflow Problem with a Separable Convex Cost Function. References. Index.

Submodular functions and optimization

The price of anarchy (POA) is a worst-case measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium. This drawback motivates the search for inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash and correlated equilibria; or to sequences of outcomes generated by natural experimentation strategies, such as successive best responses or simultaneous regret-minimization. We prove a general and fundamental connection between the price of anarchy and its seemingly stronger relatives in classes of games with a sum objective. First, we identify a "canonical sufficient condition" for an upper bound of the POA for pure Nash equilibria, which we call a smoothness argument. Second, we show that every bound derived via a smoothness argument extends automatically, with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of regret-minimizing players (or "price of total anarchy"). Smoothness arguments also have automatic implications for the inefficiency of approximate and Bayesian-Nash equilibria and, under mild additional assumptions, for bicriteria bounds and for polynomial-length best-response sequences. We also identify classes of games --- most notably, congestion games with cost functions restricted to an arbitrary fixed set --- that are tight, in the sense that smoothness arguments are guaranteed to produce an optimal worst-case upper bound on the POA, even for the smallest set of interest (pure Nash equilibria). Byproducts of our proof of this result include the first tight bounds on the POA in congestion games with non-polynomial cost functions, and the first structural characterization of atomic congestion games that are universal worst-case examples for the POA.

/pdf/intrinsic-robustness-of-the-price-of-anarchy-6n0hpoe2v7.pdf

Intrinsic robustness of the price of anarchy

The price of anarchy, defined as the ratio of the worst-case objective function value of a Nash equilibrium of a game and that of an optimal outcome, quantifies the inefficiency of selfish behavior. Remarkably good bounds on this measure are known for a wide range of application domains. However, such bounds are meaningful only if a game's participants successfully reach a Nash equilibrium. This drawback motivates inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash equilibria and correlated equilibria, and to sequences of outcomes generated by natural experimentation strategies, such as successive best responses and simultaneous regret-minimization.We establish a general and fundamental connection between the price of anarchy and its seemingly more general relatives. First, we identify a “canonical sufficient condition” for an upper bound on the price of anarchy of pure Nash equilibria, which we call a smoothness argument. Second, we prove an “extension theorem”: every bound on the price of anarchy that is derived via a smoothness argument extends automatically, with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of every outcome sequence generated by no-regret learners. Smoothness arguments also have automatic implications for the inefficiency of approximate equilibria, for bicriteria bounds, and, under additional assumptions, for polynomial-length best-response sequences. Third, we prove that in congestion games, smoothness arguments are “complete” in a proof-theoretic sense: despite their automatic generality, they are guaranteed to produce optimal worst-case upper bounds on the price of anarchy.

/pdf/intrinsic-robustness-of-the-price-of-anarchy-2ezupdog91.pdf

Intrinsic Robustness of the Price of Anarchy

Sur le probléme des courbes gauches en Topologie

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 [Math. Oper. Res., 19 (1994), pp. 390--409]. 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 [J. Appl. Ind. Math., 3 (2009), pp. 505--518], Birnbaum, Devanur, and Xiao [Proceedings of the 12th ACM Conference on Electronic Commerce, 2011, pp. 127--136]). For the latter class this resolves an open question raised by Vazirani [Math. Oper. Res., 35 (2010), pp. 458--478]. 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 wi...

A 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 ∑ij∈E Cij(fij) over feasible flows f, where on every arc ij of the network, Cij is a convex function. We give a strongly polynomial algorithm for finding an exact optimal solution for a broad class of such problems. The key characteristic of this class is that an optimal solution can be computed exactly provided its support. This includes separable convex quadratic objectives and also certain market equilibria problems: Fisher's market with linear and with spending constraint utilities. We thereby give the first strongly polynomial algorithms for separable quadratic minimum-cost flows and for Fisher's market with spending constraint utilities, settling open questions posed e.g. in [15] and in [35], respectively. The running time is O(m4 log m) for quadratic costs, O(n4+n2(m+n log n) log n) for Fisher's markets with linear utilities and O(mn3 +m2(m+n log n) log m) for spending constraint utilities.

/pdf/strongly-polynomial-algorithm-for-a-class-of-minimum-cost-2k1e7gnuw1.pdf

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

We present a new flow-type convex program describing equilibrium solutions to linear Arrow-Debreu markets. Whereas convex formulations were previously known (lNenakov and Primak 1983; Jain 2007; Cornet 1989r), our program exhibits several new features. It provides a simple necessary and sufficient condition and a concise proof of the existence and rationality of equilibria, settling an open question raised by Vazirani l2012r. As a consequence, we also obtain a simple new proof of the result in Mertens l2003r that the equilibrium prices form a convex polyhedral set.

/pdf/a-rational-convex-program-for-linear-arrow-debreu-markets-ypgsr694f0.pdf

A Rational Convex Program for Linear Arrow-Debreu Markets

We present a min-max formula for the problem of augmenting the node-connectivity of a graph by one and give a polynomial time algorithm for finding an optimal solution We also solve the minimum-cost version for node-induced cost functions

/pdf/augmenting-undirected-node-connectivity-by-one-4aan3irwzs.pdf

Augmenting Undirected Node-Connectivity by One

We give a constant-factor approximation algorithm for the asymmetric traveling salesman problem. Our approximation guarantee is analyzed with respect to the standard LP relaxation, and thus our result confirms the conjectured constant integrality gap of that relaxation. Our techniques build upon the constant-factor approximation algorithm for the special case of node-weighted metrics. Specifically, we give a generic reduction to structured instances that resemble but are more general than those arising from node-weighted metrics. For those instances, we then solve Local-Connectivity ATSP, a problem known to be equivalent (in terms of constant-factor approximation) to the asymmetric traveling salesman problem.

László A. Végh

Papers

A Strongly Polynomial Algorithm for a Class of Minimum-Cost Flow Problems with Separable Convex Objectives

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

A Rational Convex Program for Linear Arrow-Debreu Markets

Augmenting Undirected Node-Connectivity by One

A constant-factor approximation algorithm for the asymmetric traveling salesman problem