Convex Cost Function Research Articles

Image restoration using total variation with overlapping group sparsity

Image restoration is one of the most fundamental issues in imaging science. Total variation regularization is widely used in image restoration problems for its capability to preserve edges. In the literature, however, it is also well known for producing staircase artifacts. In this work we extend the total variation with overlapping group sparsity, which we previously developed for one dimension signal processing, to image restoration. A convex cost function is given and an efficient algorithm is proposed for solving the corresponding minimization problem. In the experiments, we compare our method with several state-of-the-art methods. The results illustrate the efficiency and effectiveness of the proposed method in terms of PSNR and computing time.

Multimarket Linkages, Trade and the Productivity Puzzle

AbstractThis paper examines the relationship between firms’ productivity improvement and the volume of exports, and shows that it can be sometimes negative, which seems to be an empirical puzzle. The key lies in that we simultaneously take into account intermediate retailers (i.e. vertically) and multimarket linkages (i.e. horizontally). With convex cost functions, when market conditions worsen, the manufacturer increases supply to the retailer who is larger or more efficient in trade cost.

Overbidding and overspreading in rent-seeking experiments: Cost structure and prize allocation rules

We study experimentally the effects of cost structure and prize allocation rules on the performance of rent-seeking contests. Most previous studies use a lottery prize rule and linear cost, and find both overbidding relative to the Nash equilibrium prediction and significant variation of efforts, which we term ‘overspreading.’ We investigate the effects of allocating the prize by a lottery versus sharing it proportionally, and of convex versus linear costs of effort, while holding fixed the Nash equilibrium prediction for effort. We find the share rule results in average effort closer to the Nash prediction, and lower variation of effort. Combining the share rule with a convex cost function further enhances these results. We can explain a significant amount of non-equilibrium behavior by features of the experimental design. These results contribute towards design guidelines for contests based on behavioral principles that take into account implementation features of a contest.

On convexification of range measurement based sensor and source localization problems

This paper revisits the problem of range measurement based localization of a signal source or a sensor. The major geometric difficulty of the problem comes from the non-convex structure of optimization tasks associated with range measurements, noting that the set of source locations corresponding to a certain distance measurement by a fixed point sensor is non-convex both in two and three dimensions. Differently from various recent approaches to this localization problem, all starting with a non-convex geometric minimization problem and attempting to devise methods to compensate the non-convexity effects, we suggest a geometric strategy to compose a convex minimization problem first, that is equivalent to the initial non-convex problem, at least in noise-free measurement cases. Once the convex equivalent problem is formed, a wide variety of convex minimization algorithms can be applied. The paper also suggests a gradient based localization algorithm utilizing the introduced convex cost function for localization. Furthermore, the effects of measurement noises are briefly discussed. The design, analysis, and discussions are supported by a set of numerical simulations.

Tacit Collusion in a One‐Shot Game of Price Competition with Soft Capacity Constraints

This paper analyzes price competition in the case of two firms operating under constant returns to scale with more than one production factor. Factors are chosen sequentially in a two‐stage game generating a soft capacity constraint and implying a convex short‐term cost function in the second stage of the game. We show that tacit collusion is the only predictable result of the whole game, that is, the unique payoff‐dominant pure strategy Nash equilibrium. Technically, this paper bridges the capacity constraint literature on price competition and that of the convex cost function.

Shipment Frequency of Exporters and Demand Uncertainty

This paper analyzes how firms adjust to differences in market size and demand uncertainty by changing the frequency and size of their export shipments. In our inventory model, transportation costs and optimal shipment frequency are determined on the basis of demand as well as inventory and per shipments costs. Using a cross section of detailed monthly firm-product-destination level French export data we show that, in line with the predictions of the model, firms adjust on both margins for market size. In a stochastic setting, increased demand uncertainty is associated with larger logistics costs as well as a more convex marginal cost function. Firms adjust to increased uncertainty by reducing their sales and, for a given export volume, by reducing their number of shipments and increasing their shipment size. We show that these predictions of the model are in line with patterns in the data.

Lower semicontinuity of the solution map to a parametric elliptic optimal control problem with mixed pointwise constraints

This paper studies the solution stability of a parametric optimal control problem governed by single linear elliptic equations with mixed control-state constraints and convex cost functions. By reducing the problem to a parametric programming problem and a parametric variational inequality, we obtain sufficient conditions under which the solution map to an elliptic optimal control problem is lower semicontinuous.

The Price of Mediation

Discrete Algorithms We study the relationship between correlated equilibria and Nash equilibria. In contrast to previous work focusing on the possible benefits of a benevolent mediator, we define and bound the Price of Mediation (PoM): the ratio of the social cost (or utility) of the worst correlated equilibrium to the social cost (or utility) of the worst Nash. We observe that in practice, the heuristics used for mediation are frequently non-optimal, and from an economic perspective mediators may be inept or self-interested. Recent results on computation of equilibria also motivate our work. We consider the Price of Mediation for general games with small numbers of players and pure strategies. For two player, two strategy games we give tight bounds in the non-negative cost model and the non-negative utility model. For larger games (either more players, or more pure strategies per player, or both) we show that the PoM can be arbitrary. We also have many results on symmetric congestion games (also known as load balancing games). We show that for general convex cost functions, the PoM can grow exponentially in the number of players. We prove that the PoM is one for linear costs and at most a small constant (but can be larger than one) for concave costs. For polynomial cost functions, we prove bounds on the PoM which are exponential in the degree.

Overbidding and Overspreading in Rent-Seeking Experiments: Cost Structure and Prize Allocation Rules

We study experimentally the effects of cost structure and prize allocation rules on the performance of rent-seeking contests. Most previous studies use a lottery prize rule and linear cost, and find both overbidding relative to the Nash equilibrium prediction and significant variation of efforts, which we term ‘overspreading.’ We investigate the effects of allocating the prize by a lottery versus sharing it proportionally, and of convex versus linear costs of effort, while holding fixed the Nash equilibrium prediction for effort. We find the share rule results in average effort closer to the Nash prediction, and lower variation of effort. Combining the share rule with a convex cost function further enhances these results. We can explain a significant amount of non-equilibrium behavior by features of the experimental design. These results contribute towards design guidelines for contests based on behavioral principles that take into account implementation features of a contest.

Shipment Frequency of Exporters and Demand Uncertainty

This paper analyzes how firms adjust to differences in market size and demand uncertainty by changing the frequency and size of their export shipments. In our inventory model, transportation costs and optimal shipment frequency are determined on the basis of demand as well as inventory and per shipments costs. Using a cross section of detailed monthly firm-product-destination level French export data we show that, in line with the predictions of the model, firms adjust on both margins for market size. In a stochastic setting, increased demand uncertainty is associated with larger logistics costs as well as a more convex marginal cost function. Firms adjust to increased uncertainty by reducing their sales and, for a given export volume, by reducing their number of shipments and increasing their shipment size. We show that these predictions of the model are in line with patterns in the data.

Fully Polynomial Time Approximation Schemes for Stochastic Dynamic Programs

We present a framework for obtaining fully polynomial time approximation schemes (FPTASs) for stochastic univariate dynamic programs with either convex or monotone single-period cost functions. This framework is developed through the establishment of two sets of computational rules, namely, the calculus of $K$-approximation functions and the calculus of $K$-approximation sets. Using our framework, we provide the first FPTASs for several NP-hard problems in various fields of research such as knapsack models, logistics, operations management, economics, and mathematical finance. Extensions of our framework via the use of the newly established computational rules are also discussed.

Adaptive Penalty-Based Distributed Stochastic Convex Optimization

In this work, we study the task of distributed optimization over a network of learners in which each learner possesses a convex cost function, a set of affine equality constraints, and a set of convex inequality constraints. We propose a fully-distributed adaptive diffusion algorithm based on penalty methods that allows the network to cooperatively optimize the global cost function, which is defined as the sum of the individual costs over the network, subject to all constraints. We show that when small constant step-sizes are employed, the expected distance between the optimal solution vector and that obtained at each node in the network can be made arbitrarily small. Two distinguishing features of the proposed solution relative to other related approaches is that the developed strategy does not require the use of projections and is able to adapt to and track drifts in the location of the minimizer due to changes in the constraints or in the aggregate cost itself. The proposed strategy is also able to cope with changing network topology, is robust to network disruptions, and does not require global information or rely on central processors.

Consensus Based Approach for Economic Dispatch Problem in a Smart Grid

Economic dispatch problem (EDP) is an important class of optimization problems in the smart grid, which aims at minimizing the total cost when generating certain amount of power. In this work, a novel consensus based algorithm is proposed to solve EDP in a distributed fashion. The quadratic convex cost functions are assumed in the problem formulation, and the strongly connected communication topology is sufficient for the information exchange. Unlike centralized approaches, the proposed algorithm enables generators to collectively learn the mismatch between demand and total amount of power generation. The estimated mismatch is then used as a feedback mechanism to adjust current power generation by each generator. With a tactical initial setup, eventually, all generators can automatically minimize the total cost in a collective sense.

Markov Decision Processes with Threshold Based Piecewise Linear Optimal Policies

This letter investigates the structure of the optimal policy for a class of Markov decision processes (MDPs), having convex and piecewise linear cost function. The optimal policy is proved to have a piecewise linear structure that alternates flat and constant-slope pieces, resembling a staircase with tilted rises and as many steps (thresholds) as the breakpoints of the cost function. This particular structure makes it possible to express the policy in a very compact manner, particularly suitable to be stored in low-end devices. More importantly, the threshold-based form of the optimal policy can be exploited to reduce the computational complexity of the iterative dynamic programming (DP) algorithm used to solve the problem. These results apply to a rather wide set of optimization problems, typically involving the management of a resource buffer such as the energy stored in a battery, or the packets queued in a wireless node.

Particle swarm optimization with smart inertia factor for solving non-convex economic load dispatch problems

SUMMARY This paper proposes a new approach to solve the non-convex economic dispatch problem using particle swarm optimization with smart inertia factor (PSO-SIF) algorithm in which, inertia coefficients are controlled with respect to cost function in each population. Hence, each population has unique inertia coefficient and as a result unique velocity in convergent direction for the best group solution. The new algorithm has been implemented on four test systems in different dimensions (6, 15, 20 and 40 units) with convex and non-convex cost functions. The numerical results have been compared with the results of a few PSO variants and some recently published results. The results prove the robustness and effectiveness of the proposed method and show that it could be used as a reliable tool for solving optimization problems. Copyright © 2013 John Wiley & Sons, Ltd.

HÖLDER CONTINUITY OF THE SOLUTION MAP TO AN ELLIPTIC OPTIMAL CONTROL PROBLEM WITH MIXED CONSTRAINTS

The goal of the paper is to investigate the Holder continuity of the solution map to a parametric optimal control problem which is governed by elliptic equations with mixed control-state constraints and convex cost functions. By reducing the problem to a programming problem and parametric variational inequality, we get sufficient conditions under which the solution map is Holder continuous in parameters.

Translation-invariant shrinkage/thresholding of group sparse signals

This paper addresses signal denoising when large-amplitude coefficients form clusters (groups). The L1-norm and other separable sparsity models do not capture the tendency of coefficients to cluster (group sparsity). This work develops an algorithm, called ‘overlapping group shrinkage’ (OGS), based on the minimization of a convex cost function involving a group-sparsity promoting penalty function. The groups are fully overlapping so the denoising method is translation-invariant and blocking artifacts are avoided. Based on the principle of majorization–minimization (MM), we derive a simple iterative minimization algorithm that reduces the cost function monotonically. A procedure for setting the regularization parameter, based on attenuating the noise to a specified level, is also described. The proposed approach is illustrated on speech enhancement, wherein the OGS approach is applied in the short-time Fourier transform (STFT) domain. The OGS algorithm produces denoised speech that is relatively free of musical noise.

Efficient Market Making via Convex Optimization, and a Connection to Online Learning

We propose a general framework for the design of securities markets over combinatorial or infinite state or outcome spaces. The framework enables the design of computationally efficient markets tailored to an arbitrary, yet relatively small, space of securities with bounded payoff. We prove that any market satisfying a set of intuitive conditions must price securities via a convex cost function, which is constructed via conjugate duality. Rather than deal with an exponentially large or infinite outcome space directly, our framework only requires optimization over a convex hull. By reducing the problem of automated market making to convex optimization, where many efficient algorithms exist, we arrive at a range of new polynomial-time pricing mechanisms for various problems. We demonstrate the advantages of this framework with the design of some particular markets. We also show that by relaxing the convex hull we can gain computational tractability without compromising the market institution’s bounded budget. Although our framework was designed with the goal of deriving efficient automated market makers for markets with very large outcome spaces, this framework also provides new insights into the relationship between market design and machine learning, and into the complete market setting. Using our framework, we illustrate the mathematical parallels between cost-function-based markets and online learning and establish a correspondence between cost-function-based markets and market scoring rules for complete markets.

Discussion on “A GA-API Solution for the Economic Dispatch of Generation in Power System Operation”

The commentors feel the authors of the above-named article [ibid., vol. 27, no. 1, pp. 233-242, Feb. 2012] that presents a hybrid heuristic method combining real coded genetic algorithm and special class of ant colony optimization for economic dispatch (ED) problems is commendable. The feasibility of the proposed method to solve ED is shown, by experimenting with four test systems considering 3, 6, 15, and 40 units with convex and nonconvex cost functions and also three benchmark functions. However, we would like to seek the authors' clarification regarding four particular points.

Discussion on “A GA-API Solution for the Economic Dispatch of Generation in Power System Operation”

The authors' effort in presenting a hybrid heuristic method combining real coded genetic algorithm and special class of ant colony optimization for economic dispatch (ED) problems is commendable. The feasibility of the proposed method to solve ED is shown, by experimenting with four test systems considering 3, 6, 15, and 40 units with convex and nonconvex cost functions and also three benchmark functions.