Abstract. We propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for P (X ∈ S), for a set S defined by polynomial inequalities and a random vector X defined on Ω ⊆ R n that has a given collection of up to kth-order moments. In the univariate case, we provide optimal bounds on P (X ∈ S), when the first k moments of X are given, as the solution of a semidefinite optimization problem in k + 1 dimensions. In the multivariate case, if the sets S and Ω are given by polynomial inequalities, we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n, for fixed k.We characterize the complexity of the problem of deriving tight moment inequalities. We show that it is NP-hard to find tight bounds for k ≥ 4 and Ω = R n and for k ≥ 2 and Ω = R n + , when the data in the problem is rational. For k = 1 and Ω = R n + we show that we can find tight upper bounds by solving n convex optimization problems when the set S is convex, and we provide a polynomial time algorithm when S and Ω are unions of convex sets, over which linear functions can be optimized efficiently. For the case k = 2 and Ω = R n , we present an efficient algorithm for finding tight bounds when S is a union of convex sets, over which convex quadratic functions can be optimized efficiently.
We consider the dynamic pricing problem of a monopolist firm in a market with repeated interactions, where demand is sensitive to the firm's pricing history. Consumers have memory and are prone to human decision making biases and cognitive limitations. As the firm manipulates prices, consumers form a reference price that adjusts as an anchoring standard based on price perceptions. Purchase decisions are made by assessing prices as discounts or surcharges relative to the reference price, in the spirit of prospect theory. We prove that optimal pricing policies induce a perception of monotonic prices, whereby consumers always perceive a discount, respectively surcharge, relative to their expectations. The effect is that of a skimming or penetration strategy. The firm's optimal pricing path is monotonic on the long run, but not necessarily at the introductory stage. If consumers are loss averse, we show that optimal prices converge to a constant steady state price, characterized by a simple implicit equation; otherwise the optimal policy cycles. The range of steady states is wider the more loss averse consumers are. Steady state prices decrease with the strength of the reference effect, and with customers' memory, all else equal. Offering lower prices to frequent customers may be suboptimal, however, if these are less sensitive to price changes than occasional buyers. If managers ignore such long term implications of their pricing strategy, the model indicates that they will systematically price too low and lose revenue. Our results hold under very general reference dependent demand models.
We provide an optimization framework for computing optimal upper and lower bounds on functional expectations of distributions with special properties, given moment constraints. Bertsimas and Popescu have already shown how to obtain optimal moment inequalities for arbitrary distributions via semidefinite programming. These bounds are not sharp if the underlying distributions possess additional structural properties, including symmetry, unimodality, convexity or smoothness. For convex distribution classes that are in some sense generated by an appropriate parametric family, we use conic duality to show how optimal moment bounds can be efficiently computed as semidefinite programs. In particular, we obtain generalizations of Chebyshev's inequality for symmetric and unimodal distributions, and provide numerical calculations to compare these bounds given higher order moments. We also extend these results for multivariate distributions.
W e investigate dynamic policies for allocating scarce inventory to stochastic demand for multiple fare classes, in a network environment so as to maximize total expected revenues. Typical applications include sequential reservations for an airline network, hotel, or car rental service. We propose and analyze a new algorithm based on approximate dynamic programming, both theoretically and computationally. This algorithm uses adaptive, nonadditive bid prices from a linear programming relaxation. We provide computational results that give insight into the performance of the new algorithm and the widely used bid-price control, for several networks and demand scenarios. We extend the proposed algorithm to handle cancellations and no-shows by incorporating oversales decisions in the underlying linear programming formulation. We report encouraging computational results that show that the new algorithm leads to higher revenues and more robust performance than bid-price control.
We provide a method for deriving robust solutions to certain stochastic optimization problems, based on mean-covariance information about the distributions underlying the uncertain vector of returns. We prove that for a general class of objective functions, the robust solutions amount to solving a certain deterministic parametric quadratic program. We first prove a general projection property for multivariate distributions with given means and covariances, which reduces our problem to optimizing a univariate mean-variance robust objective. This allows us to use known univariate results in the multidimensional setting, and to add new results in this direction. In particular, we characterize a general class of objective functions (the so-called one- or two-point support functions), for which the robust objective is reduced to a deterministic optimization problem in one variable. Finally, we adapt a result from Geoffrion (1967a) to reduce the main problem to a parametric quadratic program. In particular, our results are true for increasing concave utilities with convex or concave-convex derivatives. Closed-form solutions are obtained for special discontinuous criteria, motivated by bonus- and commission-based incentive schemes for portfolio management. We also investigate a multiproduct pricing application, which motivates extensions of our results for the case of nonnegative and decision-dependent returns.
We analyze a dynamic pricing problem where consumer's purchase decisions are affected by representative past prices, summarized in a reference price. We propose a new, behaviorally motivated reference price mechanism, based on the peak-end memory model proposed by Fredrickson and Kahneman (1993). Specifically, we assume that consumers' reference price is a weighted average of the lowest and last price. Gain or loss perceptions with respect to this reference price affect consumer purchase decisions in the spirit of prospect theory, resulting in a non-smooth demand function. We investigate how these behavioral patterns in consumer anchoring and decision processes affect the optimal dynamic pricing policy of the firm. In contrast with previous literature, we show that peak-end anchoring leads to a range of optimal constant pricing policies even with loss neutral buyers. This range becomes wider if consumers are loss averse. In general, we show that skimming or penetration strategies are optimal, i.e. the transient pricing policy is monotone, and converges to a steady state, which depends on the initial price perception. The value of the steady state price decreases, the more consumers are sensitive to price changes, and the more they anchor on the lowest price.
The idea of investigating the relation of option and stock prices just based on the noarbitrage assumption, but without assuming any model for the underlying price dynamics has a long history in the financial economics literature. We introduce convex, and in particular semidefinite, optimization methods, duality and complexity theory to shed new light to this relation. For the single stock problem, given moments of the prices of the underlying assets, we show that we can find best possible bounds on option prices with general payoff functions efficiently, either algorithmically (solving a semidefinite optimization problem) or in closed form. Conversely, given observable option prices, we provide best possible bounds on moments of the prices of the underlying assets as well as on the prices of other options on the same asset by solving linear optimization problems. For options that are affected by multiple stocks either directly (the payoff of the option depends on multiple stocks) or indirectly (we have information on correlations between stock prices), we find bounds (but not best possible ones) using convex optimization methods. However, we show it is NP-hard to find best possible bounds in multiple dimensions. We extend our results under transactions costs as well.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.