Non-linear pricing by convex duality

We consider the pricing problem of a risk-neutral monopolist who produces (at a cost) and offers an infinitely divisible good to a single potential buyer that can be of a finite number of (single dimensional) types. The buyer has a non-linear utility function that is differentiable, strictly concave and strictly increasing. Using a simple reformulation and shortest path problem duality as in Vohra (2011) we transform the initial non-convex pricing problem of the monopolist into an equivalent optimization problem yielding a closed-form pricing formula under a regularity assumption on the probability distribution of buyer types. We examine the solution of the problem when the regularity condition is relaxed in different ways, or when the production function is non-linear and convex. For arbitrary type distributions, we offer a complete solution procedure. © 2015 Elsevier Ltd. All rights reserved

Generalized second price auction is optimal for discrete types

We prove that a variant of the second price auction for the sale of a single good through a Bayesian incentive compatible mechanism that maximizes expected revenue of the seller is optimal when the type space is discrete. Moreover, we show that this variant is related to the widely used generalized second price auction mechanism in keyword-auctions for advertising, thus providing a theoretical justification for a practical tool. © 2016 Elsevier B.V.

Huber approximation for the non-linear ℓ1 problem

The smooth Huber approximation to the non-linear ℓ1 problem was proposed by Tishler and Zang (1982), and further developed in Yang (1995). In the present paper, we use the ideas of Gould (1989) to give a new algorithm with rate of convergence results for the smooth Huber approximation. Results of computational tests are reported. © 2005 Elsevier B.V. All rights reserved

A penalty continuation method for the ℓ∞ solution of overdetermined linear systems

A new algorithm for the ℓ∞ solution of overdetermined linear systems is given. The algorithm is based on the application of quadratic penalty functions to a primal linear programming formulation of the ℓ∞ problem. The minimizers of the quadratic penalty function generate piecewise-linear non-interior paths to the set of ℓ∞ solutions. It is shown that the entire set of ℓ∞ solutions is obtained from the paths for sufficiently small values of a scalar parameter. This leads to a finite penalty/continuation algorithm for ℓ∞ problems. The algorithm is implemented and extensively tested using random and function approximation problems. Comparisons with the Barrodale-Phillips simplex based algorithm and the more recent predictor-corrector primal-dual interior point algorithm are given. The results indicate that the new algorithm shows a promising performance on random (non-function approximation) problems

A bilevel uncapacitated location/pricing problem with Hotelling access costs in one-dimensional space

We formulate a spatial pricing problem as bilevel non-capacitated location: A leader first decides which facilities to open and sets service prices taking competing offers into account; then, customers make individual decisions minimizing individual costs that include access charges in the spirit of Hotelling. Both leader and customers are assumed to be risk-neutral. For non-metric costs (i.e., when access costs do not satisfy the triangle inequality), the problem is NP-hard even if facilities can be opened at no fixed cost. We describe an algorithm for solving the Euclidean 1-dimensional case (i.e., with access cost defined by the Euclidean norm on a line) with fixed opening costs and a single competing facility

A finite continuation algorithm for bound constrained quadratic programming

The dual of the strictly convex quadratic programming problem with unit bounds is posed as a linear ℓ1 minimization problem with quadratic terms. A smooth approximation to the linear ℓ1 function is used to obtain a parametric family of piecewise-quadratic approximation problems. The unique path generated by the minimizers of these problems yields the solution to the original problem for finite values of the approximation parameter. Thus, a finite continuation algorithm is designed. Results of extensive computational experiments are reported

The robust network loading problem under hose demand uncertainty: Formulation, polyhedral analysis, and computations

We consider the network loading problem (NLP) under a polyhedral uncertainty description of traffic demands. After giving a compact multicommodity flow formulation of the problem, we state a decomposition property obtained from projecting out the flow variables. This property considerably simplifies the resulting polyhedral analysis and computations by doing away with metric inequalities. Then we focus on a specific choice of the uncertainty description, called the "hose model," which specifies aggregate traffic upper bounds for selected endpoints of the network. We study the polyhedral aspects of the NLP under hose demand uncertainty and use the results as the basis of an efficient branch-and-cut algorithm. The results of extensive computational experiments on well-known network design instances are reported. © 2011 INFORMS

A model and case study for efficient shelf usage and assortment analysis

In the rapidly changing environment of Fast Moving Consumer Goods sector where new product launches are frequent, retail channels need to reallocate their shelf spaces intelligently while keeping up their total profit margins, and to simultaneously avoid product pollution. In this paper we propose an optimization model which yields the optimal product mix on the shelf in terms of profitability, and thus helps the retailers to use their shelves more effectively. The model is applied to the shampoo product class at two regional supermarket chains. The results reveal not only a computationally viable model, but also substantial potential increases in the profitability after the reorganization of the product list. © 2008 Springer Science+Business Media, LLC

Constrained nonlinear programming for volatility estimation with GARCH models

This paper proposes a constrained nonlinear programming view of generalized autoregressive conditional heteroskedasticity (GARCH) volatility estimation models in financial econometrics. These models are usually presented to the reader as unconstrained optimization models with recursive terms in the literature, whereas they actually fall into the domain of nonconvex nonlinear programming. Our results demonstrate that constrained nonlinear programming is a worthwhile exercise for GARCH models, especially for the bivariate and trivariate cases, as they offer a significant improvement in the quality of the solution of the optimization problem over the diagonal VECH and the BEKK representations of the multivariate GARCH model

Structured least squares with bounded data uncertainties

In many signal processing applications the core problem reduces to a linear system of equations. Coefficient matrix uncertainties create a significant challenge in obtaining reliable solutions. In this paper, we present a novel formulation for solving a system of noise contaminated linear equations while preserving the structure of the coefficient matrix. The proposed method has advantages over the known Structured Total Least Squares (STLS) techniques in utilizing additional information about the uncertainties and robustness in ill-posed problems. Numerical comparisons are given to illustrate these advantages in two applications: signal restoration problem with an uncertain model and frequency estimation of multiple sinusoids embedded in white noise. ©2009 IEEE