A Decomposition Algorithm for Nested Resource Allocation Problems

We propose an exact polynomial algorithm for a resource allocation problem with convex costs and constraints on partial sums of resource consumptions, in the presence of either continuous or integer variables. No assumption of strict convexity or differentiability is needed. The method solves a hierarchy of resource allocation subproblems, whose solutions are used to convert constraints on sums of resources into bounds for separate variables at higher levels. The resulting time complexity for the integer problem is $O(n \log m \log (B/n))$, and the complexity of obtaining an $\epsilon$-approximate solution for the continuous case is $O(n \log m \log (B/\epsilon))$, $n$ being the number of variables, $m$ the number of ascending constraints (such that $m < n$), $\epsilon$ a desired precision, and $B$ the total resource. This algorithm attains the best-known complexity when $m = n$, and improves it when $\log m = o(\log n)$. Extensive experimental analyses are conducted with four recent algorithms on various continuous problems issued from theory and practice. The proposed method achieves a higher performance than previous algorithms, addressing all problems with up to one million variables in less than one minute on a modern computer.Comment: Working Paper -- MIT, 23 page

On Solving Convex Optimization Problems with Linear Ascending Constraints

In this paper, we propose two algorithms for solving convex optimization problems with linear ascending constraints. When the objective function is separable, we propose a dual method which terminates in a finite number of iterations. In particular, the worst case complexity of our dual method improves over the best-known result for this problem in Padakandla and Sundaresan [SIAM J. Optimization, 20 (2009), pp. 1185-1204]. We then propose a gradient projection method to solve a more general class of problems in which the objective function is not necessarily separable. Numerical experiments show that both our algorithms work well in test problems.Comment: 20 pages. The final version of this paper is published in Optimization Letter

Designing Coalition-Proof Reverse Auctions over Continuous Goods

This paper investigates reverse auctions that involve continuous values of different types of goods, general nonconvex constraints, and second stage costs. We seek to design the payment rules and conditions under which coalitions of participants cannot influence the auction outcome in order to obtain higher collective utility. Under the incentive-compatible Vickrey-Clarke-Groves mechanism, we show that coalition-proof outcomes are achieved if the submitted bids are convex and the constraint sets are of a polymatroid-type. These conditions, however, do not capture the complexity of the general class of reverse auctions under consideration. By relaxing the property of incentive-compatibility, we investigate further payment rules that are coalition-proof without any extra conditions on the submitted bids and the constraint sets. Since calculating the payments directly for these mechanisms is computationally difficult for auctions involving many participants, we present two computationally efficient methods. Our results are verified with several case studies based on electricity market data

MECHANISM DESIGN WITH GENERAL UTILITIES

This thesis studies mechanism design from an optimization perspective. Our main contribution is to characterize fundamental structural properties of optimization problems arising in mechanism design and to exploit them to design general frameworks and techniques for efficiently solving the underlying problems. Not only do our characterizations allow for efficient computation, they also reveal qualitative characteristics of optimal mechanisms which are important even from a non-computational standpoint. Furthermore, most of our techniques are widely applicable to optimization problems outside of mechanism design such as online algorithms or stochastic optimization. Our frameworks can be summarized as follows. When the input to an optimization problem (e.g., a mechanism design problem) comes from independent sources (e.g., independent agents), the complexity of the problem can be exponentially reduced by (i) decomposing the problem into smaller subproblems, each one involving one input source, (ii) simultaneously optimizing the subproblems subject to certain relaxation of coupling constraints, and (iii) combining the solutions of the subproblems in a certain way to obtain an (approximately) optimal solution for the original problem. We use our proposed framework to construct optimal or approximately optimal mechanisms for several settings previously considered in the literature and to improve upon the best previously known results. We also present applications of our techniques to non-mechanism design problems such as online stochastic generalized assignment problem which itself captures online and stochastic versions of various other problems such as resource allocation and job scheduling

On a reduction for a class of resource allocation problems

In the resource allocation problem (RAP), the goal is to divide a given amount of resource over a set of activities while minimizing the cost of this allocation and possibly satisfying constraints on allocations to subsets of the activities. Most solution approaches for the RAP and its extensions allow each activity to have its own cost function. However, in many applications, often the structure of the objective function is the same for each activity and the difference between the cost functions lies in different parameter choices such as, e.g., the multiplicative factors. In this article, we introduce a new class of objective functions that captures the majority of the objectives occurring in studied applications. These objectives are characterized by a shared structure of the cost function depending on two input parameters. We show that, given the two input parameters, there exists a solution to the RAP that is optimal for any choice of the shared structure. As a consequence, this problem reduces to the quadratic RAP, making available the vast amount of solution approaches and algorithms for the latter problem. We show the impact of our reduction result on several applications and, in particular, we improve the best known worst-case complexity bound of two important problems in vessel routing and processor scheduling from $O(n^2)$ to $O(n \log n)$

Do Prices Coordinate Markets?

Walrasian equilibrium prices can be said to coordinate markets: They support a welfare optimal allocation in which each buyer is buying bundle of goods that is individually most preferred. However, this clean story has two caveats. First, the prices alone are not sufficient to coordinate the market, and buyers may need to select among their most preferred bundles in a coordinated way to find a feasible allocation. Second, we don't in practice expect to encounter exact equilibrium prices tailored to the market, but instead only approximate prices, somehow encoding "distributional" information about the market. How well do prices work to coordinate markets when tie-breaking is not coordinated, and they encode only distributional information? We answer this question. First, we provide a genericity condition such that for buyers with Matroid Based Valuations, overdemand with respect to equilibrium prices is at most 1, independent of the supply of goods, even when tie-breaking is done in an uncoordinated fashion. Second, we provide learning-theoretic results that show that such prices are robust to changing the buyers in the market, so long as all buyers are sampled from the same (unknown) distribution

Designing Matching Mechanisms under Constraints: An Approach from Discrete Convex Analysis

We consider two-sided matching problems where agents on one side of the market (hospitals) are required to satisfy certain distributional constraints. We show that when the preferences and constraints of the hospitals can be represented by an M-natural-concave function, (i) the generalized Deferred Acceptance mechanism is strategyproof for doctors, (ii) it produces the doctor-optimal stable matching, and (iii) its time complexity is proportional to the square of the number of possible contracts. Furthermore, we provide sufficient conditions for representation by an M-natural-concave function. These conditions are applicable to various existing works and enable new applications as well, thereby providing a recipe for developing desirable mechanisms in practice

Nash welfare, valuated matroids, and gross substitutes

We study computational aspects of equilibria and fair division problems with a focus on demand and valuation functions that satisfy the (weak) gross substitutes property. We study the Arrow-Debreu exchange market model with divisible goods where agents’ demands satisfy the weak gross substitutes (WGS) property. We give an auction algorithm that obtains an approximate market equilibrium for WGS demands. Previously, such algorithms were known only for restricted classes of WGS demands. We also derive the implications of our technique for spending-restricted market equilibrium for budget-separable piecewise linear concave (budget-SPLC) utilities. Spending-restricted equilibrium was introduced as a continuous relaxation of the Nash SocialWelfare (NSW) problem. Next, we present the first polynomial-time constant-factor approximation algorithm for the NSW problem under Rado valuations. Rado valuations form a general class of valuation functions that arise from maximum cost independent matching problems. They include as special cases assignment (OXS) valuations and weighted matroid rank functions. Our approach also gives the first polynomial-time constant-factor approximation algorithm for the asymmetric NSW problem under Rado valuations, provided that the maximum ratio between the weights is bounded by a constant. We examine the Matroid Based Valuation (MBV) conjecture by Ostrovsky and Paes Leme (Theoretical Economics 2015). It asserts that every (discrete) gross substitute valuation is a matroid based valuation—a valuation obtained from weighted matroid rank functions by repeated applications of merge and endowment operations. Each matroid based valuation turns out to be an endowment of some Rado valuation. By introducing complete classes of valuated matroids, we exhibit a family of valuations that are gross substitutes but not endowed Rado valuations. This refutes the MBV conjecture. The family is defined via sparse paving matroids