Showing papers on "Concave function published in 2010"

Tractable robust expected utility and risk models for portfolio optimization

Abstract: Expected utility models in portfolio optimization are based on the assumption of complete knowledge of the distribution of random returns. In this paper, we relax this assumption to the knowledge of only the mean, covariance, and support information. No additional restrictions on the type of distribution such as normality is made. The investor’s utility is modeled as a piecewise-linear concave function. We derive exact and approximate optimal trading strategies for a robust (maximin) expected utility model, where the investor maximizes his worst-case expected utility over a set of ambiguous distributions. The optimal portfolios are identified using a tractable conic programming approach. Extensions of the model to capture asymmetry using partitioned statistics information and box-type uncertainty in the mean and covariance matrix are provided. Using the optimized certainty equivalent framework, we provide connections of our results with robust or ambiguous convex risk measures, in which the investor minimizes his worst-case risk under distributional ambiguity. New closed-form results for the worst-case optimized certainty equivalent risk measures and optimal portfolios are provided for two- and three-piece utility functions. For more complicated utility functions, computational experiments indicate that such robust approaches can provide good trading strategies in financial markets.

Efficient Minimization of Decomposable Submodular Functions

Abstract: Many combinatorial problems arising in machine learning can be reduced to the problem of minimizing a submodular function. Submodular functions are a natural discrete analog of convex functions, and can be minimized in strongly polynomial time. Unfortunately, state-of-the-art algorithms for general submodular minimization are intractable for larger problems. In this paper, we introduce a novel subclass of submodular minimization problems that we call decomposable. Decomposable submodular functions are those that can be represented as sums of concave functions applied to modular functions. We develop an algorithm, SLG, that can efficiently minimize decomposable submodular functions with tens of thousands of variables. Our algorithm exploits recent results in smoothed convex minimization. We apply SLG to synthetic benchmarks and a joint classification-and-segmentation task, and show that it outperforms the state-of-the-art general purpose submodular minimization algorithms by several orders of magnitude.

Efficient Minimization of Decomposable Submodular Functions

Abstract: Many combinatorial problems arising in machine learning can be reduced to the problem of minimizing a submodular function. Submodular functions are a natural discrete analog of convex functions, and can be minimized in strongly polynomial time. Unfortunately, state-of-the-art algorithms for general submodular minimization are intractable for larger problems. In this paper, we introduce a novel subclass of submodular minimization problems that we call decomposable. Decomposable submodular functions are those that can be represented as sums of concave functions applied to modular functions. We develop an algorithm, SLG, that can efficiently minimize decomposable submodular functions with tens of thousands of variables. Our algorithm exploits recent results in smoothed convex minimization. We apply SLG to synthetic benchmarks and a joint classification-and-segmentation task, and show that it outperforms the state-of-the-art general purpose submodular minimization algorithms by several orders of magnitude.

Dynamic Optimization and Learning for Renewal Systems

Abstract: We consider the problem of optimizing time averages in systems with independent and identically distributed behavior over renewal frames. This includes scheduling and task processing to maximize utility in stochastic networks with variable length scheduling modes. Every frame, a new policy is implemented that affects the frame size and that creates a vector of attributes. An algorithm is developed for choosing policies on each frame in order to maximize a concave function of the time average attribute vector, subject to additional time average constraints. The algorithm is based on Lyapunov optimization concepts and involves minimizing a ``drift-plus-penalty'' ratio over each frame. The algorithm can learn efficient behavior without a-priori statistical knowledge by sampling from the past. Our framework is applicable to a large class of problems, including Markov decision problems.

On the One-Dimensional Optimal Switching Problem

Abstract: We explicitly solve the optimal switching problem for one-dimensional diffusions by directly using the dynamic programming principle and the excessive characterization of the value function. The shape of the value function and the smooth fit principle then can be proved using the properties of concave functions.

The complexity of equilibria in cost sharing games

Abstract: We study Congestion Games with non-increasing cost functions (Cost Sharing Games) from a complexity perspective and resolve their computational hardness, which has been an open question. Specifically we prove that when the cost functions have the form f(x) = cr/x (Fair Cost Allocation) then it is PLS-complete to compute a Pure Nash Equilibrium even in the case where strategies of the players are paths on a directed network. For cost functions of the form f(x) = cr(x)/x, where cr(x) is a non-decreasing concave function we also prove PLScompleteness in undirected networks. Thus we extend the results of [7, 1] to the non-increasing case. For the case of Matroid Cost Sharing Games, where tractability of Pure Nash Equilibria is known by [1] we give a greedy polynomial time algorithm that computes a Pure Nash Equilibrium with social cost at most the potential of the optimal strategy profile. Hence, for this class of games we give a polynomial time version of the Potential Method introduced in [2] for bounding the Price of Stability.

Optimizing power allocation in interference channels using D.C. programming

Abstract: Power allocation is a promising approach for optimizing the performance of mobile radio systems in interference channels. In the present paper, the non-convex objective function of the power allocation problem aiming at maximizing the sum rate with a total power constraint is reformulated as a difference of two concave functions. A global optimum power allocation is found by applying a branch and bound based algorithm to the new formulation. The algorithm basically splits the feasible region consecutively into subregions where for every subregion the objective function is upper and lower bounded. For a certain partition of the feasible region, a power allocation corresponding to the highest lower bound which is upper bounded by the highest upper bound with some insignificant difference is found as the global optimum. A convex maximization formulation of the optimization problem with a piecewise linearly outer approximated feasible region is essentially applied for finding an upper bound which only requires solving a linear program problem. The simulation results show a significant improvement in the sum rate of the proposed algorithm over the conventional suboptimal techniques.

Approximation and Shape Preserving Properties of the Nonlinear Meyer–König and Zeller Operator of Max-Product Kind

Abstract: Starting from the study of the Shepard nonlinear operator of max-prod type in [2, 3; 6, Open Problem 5.5.4], the Meyer–Konig and Zeller max-product type operator is introduced and the question of the approximation order by this operator is raised. The first aim of this article is to obtain the order of pointwise approximation for these operators. Also, we prove by a counterexample that in some sense, in general this type of order of approximation with respect to ω1(f; ·) cannot be improved. However, for some subclasses of functions, including for example the continuous nondecreasing concave functions, the essentially better order (of uniform approximation) ω1(f; 1/n) is obtained. Several shape preserving properties are obtained including the preservation of quasi-convexity.

Jensen's Inequality, Parameter Uncertainty, and Multi-Period Investment

Abstract: Classical approaches to estimation and decisions requiring estimation often are at odds. When values critical to the decision are convex or concave functions of unknown parameters, the statistician’s estimation error adjustments are the opposite of what is appropriate for the decision. We illustrate the conflict by studying multi-period investment problems. The proper application of Jensen’s inequality to the decision turns finance intuition on its head. For example, multi-period investments with negative risk premia can be profitable, there can be infinite demand for risky securities by risk averse investors, settings exist where risk averse investors should not diversify, and demand for mutual funds with negative alphas may be rational.

Dynamic optimization and learning for renewal systems

Abstract: We consider the problem of optimizing time averages in systems with independent and identically distributed behavior over renewal frames. This includes scheduling and task processing to maximize utility in stochastic networks with variable length scheduling modes. Every frame, a new policy is implemented that affects the frame size and that creates a vector of attributes. An algorithm is developed for choosing policies on each frame in order to maximize a concave function of the time average attribute vector, subject to additional time average constraints. The algorithm is based on Lyapunov optimization concepts and involves minimizing a “drift-plus-penalty” ratio over each frame. The algorithm can learn efficient behavior without a-priori statistical knowledge by sampling from the past. Our framework is applicable to a large class of problems, including Markov decision problems.

Optimal smoothing spline with constraints on its derivatives

Abstract: This paper considers the problem of designing optimal smoothing spline with constraints on its derivatives. The splines of degree k are constituted by employing normalized uniform B-splines as the basis functions. We then show that the l-th derivative of the spline can be obtained by using B-splines of degree k-l with the control points computed as l-th difference of original control points. This yields systematic treatment of equality and inequality constraints over intervals on derivatives of arbitrary degree. Also, pointwise constraints can readily be incorporated. The problem of optimal smoothing splines with constraints reduce to convex quadratic programming problems. The effectiveness is demonstrated by numerical examples of approximations of probability distribution function and concave function, and trajectory planning with the constraints on velocity and acceleration.

Contribution games in social networks

Abstract: We consider network contribution games, where each agent in a social network has a budget of effort that he can contribute to different collaborative projects. Depending on the contribution of the involved agents a project will be successful to a different degree, and to measure the success we use a reward function for each project. Every agent is trying to maximize the reward from all projects that it is involved in. We consider pairwise equilibria of this game and characterize the existence, computational complexity, and quality of equilibrium based on the types of reward functions involved. For example, when all reward functions are concave, we prove that the price of anarchy is at most 2. For convex functions the same only holds under some special but very natural conditions. A special focus of the paper are minimum effort games, where the success of a project depends only on the minimum effort of any of the participants. Finally, we briefly discuss additional aspects like approximate equilibria and convergence of dynamics.

Counting in Graph Covers: A Combinatorial Characterization of the Bethe Entropy Function

Abstract: We present a combinatorial characterization of the Bethe entropy function of a factor graph, such a characterization being in contrast to the original, analytical, definition of this function. We achieve this combinatorial characterization by counting valid configurations in finite graph covers of the factor graph. Analogously, we give a combinatorial characterization of the Bethe partition function, whose original definition was also of an analytical nature. As we point out, our approach has similarities to the replica method, but also stark differences. The above findings are a natural backdrop for introducing a decoder for graph-based codes that we will call symbolwise graph-cover decoding, a decoder that extends our earlier work on blockwise graph-cover decoding. Both graph-cover decoders are theoretical tools that help towards a better understanding of message-passing iterative decoding, namely blockwise graph-cover decoding links max-product (min-sum) algorithm decoding with linear programming decoding, and symbolwise graph-cover decoding links sum-product algorithm decoding with Bethe free energy function minimization at temperature one. In contrast to the Gibbs entropy function, which is a concave function, the Bethe entropy function is in general not concave everywhere. In particular, we show that every code picked from an ensemble of regular low-density parity-check codes with minimum Hamming distance growing (with high probability) linearly with the block length has a Bethe entropy function that is convex in certain regions of its domain.

Base‐stock policies in capacitated assembly systems: Convexity properties

Abstract: We study an assembly system with a single finished product managed using an echelon base-stock or order-up-to policy. Some or all operations have capacity constraints. Excess demand is either backordered in every period or lost in every period. We show that the shortage penalty cost over any horizon is jointly convex with respect to the base-stock levels and capacity levels. When the holding costs are also included in the objective function, we show that the cost function can be written as a sum of a convex function and a concave function. Throughout the article, we discuss algorithmic implications of our results for making optimal inventory and capacity decisions in such systems.© 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2010

The VPN Problem with Concave Costs

Abstract: Only recently Goyal, Olver, and Shepherd [Proc. STOC, ACM, New York, 2008] proved that the symmetric virtual private network design (sVPN) problem has the tree routing property, namely, that there always exists an optimal solution to the problem whose support is a tree. Combining this with previous results by Fingerhut, Suri, and Turner [J. Algorithms, 24 (1997), pp. 287-309] and Gupta et al. [Proc. STOC, ACM, New York, 2001], sVPN can be solved in polynomial time. In this paper we investigate an APX-hard generalization of sVPN, where the contribution of each edge to the total cost is proportional to some non-negative, concave, and nondecreasing function of the capacity reservation. We show that the tree routing property extends to the new problem and give a constant-factor approximation algorithm for it. We also show that the undirected uncapacitated single-source minimum concave-cost flow problem has the tree routing property when the cost function has some property of symmetry.

Global Optimality Conditions for Some Classes of Optimization Problems

Abstract: We establish new necessary and sufficient optimality conditions for global optimization problems. In particular, we establish tractable optimality conditions for the problems of minimizing a weakly convex or concave function subject to standard constraints, such as box constraints, binary constraints, and simplex constraints. We also derive some new necessary and sufficient optimality conditions for quadratic optimization. Our main theoretical tool for establishing these optimality conditions is abstract convexity.

Parallel machine scheduling problems considering regular measures of performance and machine cost

Abstract: This research considers a broad range of scheduling problems in the parallel machines environment. Schedules are evaluated according to two independent components of the objective function: (1) machine cost consisting of a fixed cost and a variable cost; and (2) a regular measure of performance. This study is only one of a few that take the selection of machines among those available as a decision variable. For machine cost with concave functions, we derive the general characteristics of optimal solutions with respect to decisions on the number of machines to use and the way to load the machines. Our analysis is not restricted to the machine cost criterion, but may be extended to other measures with concave functions. Furthermore, we provide a Pareto efficient perspective in understanding the tradeoff between machine cost and any regular measure of performance.

Might randomization in queue discipline be useful when waiting cost is a concave function of waiting time

Abstract: This paper suggests that introducing randomization in queue discipline might be welfare enhancing in certain queues for which the cost of waiting is a concave function of waiting time. Concavity can make increased variability in waiting times good not bad for aggregate customer welfare. Such concavity may occur if the costs of waiting asymptotically approach some maximum or if the customer incurs a fixed cost if there is any wait at all. As examples, cost might asymptotically approach a maximum for patients seeking organ transplants who will not live beyond a certain threshold time, and fixed costs could pertain for knowledge workers seeking a piece of information that is required to proceed with their current task, so any delay creates a “set up charge” associated with switching tasks.

One Tree Suffices: A Simultaneous O(1)-Approximation for Single-Sink Buy-at-Bulk

Abstract: We study the single-sink buy-at-bulk problem with an unknown cost function. We wish to route flow from a set of demand nodes to a root node, where the cost of routing x total flow along an edge is proportional to f(x) for some concave, non-decreasing function f satisfying f(0)=0. We present a simple, fast, combinatorial algorithm that takes a set of demands and constructs a single tree T such that for all f the cost f(T) is a 47.45-approximation of the optimal cost for that f. This is within a factor of 2.33 of the best approximation ratio currently achievable when the tree can be optimized for a specific function. Trees achieving simultaneous O(1)-approximations for all concave functions were previously not known to exist regardless of computation time.

The geometric convexity of a function involving gamma function with applications

Abstract: In this paper, we prove that is geometrically convex on (0, ). As its applications, we obtain some new estimates for .

One Tree Suffices: A Simultaneous O(1)-Approximation for Single-Sink Buy-at-Bulk

Abstract: We study the single-sink buy-at-bulk problem with an unknown cost function. We wish to route flow from a set of demand nodes to a root node, where the cost of routing x total flow along an edge is proportional to f(x) for some concave, non-decreasing function f satisfying f(0)=0. We present a simple, fast, combinatorial algorithm that takes a set of demands and constructs a single tree T such that for all f the cost f(T) is a 47.45-approximation of the optimal cost for that f. This is within a factor of 2.33 of the best approximation ratio currently achievable when the tree can be optimized for a specific function. Trees achieving simultaneous O(1)-approximations for all concave functions were previously not known to exist regardless of computation time.

Optimal multicommodity flow through the complete graph with random edge capacities

Abstract: We consider a multicommodity flow problem on a complete graph whose edges have random, independent, and identically distributed capacities. We show that, as the number of nodes tends to infinity, the maximumutility, given by the average of a concave function of each commodity How, has an almost-sure limit. Furthermore, the asymptotically optimal flow uses only direct and two-hop paths, and can be obtained in a distributed manner.