Concave function

About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.

Supply function equilibrium in electricity spot markets with contracts and price caps

Abstract: In electricity wholesale markets, generators often sign long term contracts with purchasers of power in order to hedge risks. In this paper, we consider a market where demand is uncertain, but can be represented as a function of price together with a random shock. Each generator offers a smooth supply function into the market and wishes to maximize his expected profit, allowing for his contract position. We investigate supply function equilibria in this setting, using a model introduced by Anderson and Philpott. We study first the existence of a unique monotonically increasing supply curve that maximizes the objective function under the constraint of limited generation capacity and a price cap, and discuss the influence of the generator’s contract on the optimal supply curve. We then investigate the existence of a symmetric Nash supply function equilibrium, where we do not have to assume that the demand is a concave function of price. Finally, we identify the Nash supply function equilibrium which gives rise to the generators’ maximal expected profit.

Utility Maximization with a Given Pricing Measure When the Utility Is Not Necessarily Concave

Abstract: We study the problem of maximizing expected utility from terminal wealth for a non-concave utility function and for a budget set given by one fixed pricing measure. We prove the existence and several fundamental properties of a maximizer. We analyze the (non-concave) value function (indirect utility). In particular, we show that the concave envelope of the non-concave value function is the value function of the utility maximization problem for the concave envelope of the non-concave utility function. The two value functions are shown to coincide if the underlying probability space is atomless. This allows us to characterize the maximizers for several model classes explicitly.

Optimal Battery Dimensioning and Control of a CVT PHEV Powertrain

Abstract: This paper presents convex modeling steps for the problem of optimal battery dimensioning and control of a plug-in hybrid electric vehicle with a continuous variable transmission. The power limits of the internal combustion engine and the electric machine are approximated as convex/concave functions in kinetic energy, whereas their losses are approximated as convex in both kinetic energy and power. An example of minimizing the total cost of ownership of a city bus including a battery wear model is presented. The proposed method is also used to obtain optimal charging power from an infrastructure that is to be designed at the same time the bus is dimensioned.

Capacity upper bounds for deletion-type channels

Abstract: We develop a systematic approach, based on convex programming and real analysis, for obtaining upper bounds on the capacity of the binary deletion channel and, more generally, channels with i.i.d. insertions and deletions. Other than the classical deletion channel, we give a special attention to the Poisson-repeat channel introduced by Mitzenmacher and Drinea (IEEE Transactions on Information Theory, 2006). Our framework can be applied to obtain capacity upper bounds for any repetition distribution (the deletion and Poisson-repeat channels corresponding to the special cases of Bernoulli and Poisson distributions). Our techniques essentially reduce the task of proving capacity upper bounds to maximizing a univariate, real-valued, and often concave function over a bounded interval. The corresponding univariate function is carefully designed according to the underlying distribution of repetitions and the choices vary depending on the desired strength of the upper bounds as well as the desired simplicity of the function (e.g., being only efficiently computable versus having an explicit closed-form expression in terms of elementary, or common special, functions). Among our results, we show that the capacity of the binary deletion channel with deletion probability d is at most (1−d) logϕ for d ≥ 1/2, and, assuming the capacity function is convex, is at most 1−d log(4/ϕ) for dd outside the limiting case d → 0 that is fully explicit and proved without computer assistance. Furthermore, we derive the first set of capacity upper bounds for the Poisson-repeat channel. Our results uncover further striking connections between this channel and the deletion channel, and suggest, somewhat counter-intuitively, that the Poisson-repeat channel is actually analytically simpler than the deletion channel and may be of key importance to a complete understanding of the deletion channel. Finally, we derive several novel upper bounds on the capacity of the deletion channel. All upper bounds are maximums of efficiently computable, and concave, univariate real functions over a bounded domain. In turn, we upper bound these functions in terms of explicit elementary and standard special functions, whose maximums can be found even more efficiently (and sometimes, analytically, for example for d=1/2). Along the way, we develop several new techniques of potentially independent interest. For example, we develop systematic techniques to study channels with mean constraints over the reals. Furthermore, we motivate the study of novel probability distributions over non-negative integers, as well as novel special functions which could be of interest to mathematical analysis.