Concave function

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

Optimal Control with State Constraint and Non-concave Dynamics: A Model Arising in Economic Growth

Abstract: We consider an optimal control problem arising in the context of economic theory of growth, on the lines of the works by Skiba and Askenazy–Le Van. The framework of the model is intertemporal infinite horizon utility maximization. The dynamics involves a state variable representing total endowment of the social planner or average capital of the representative dynasty. From the mathematical viewpoint, the main features of the model are the following: (i) the dynamics is an increasing, unbounded and not globally concave function of the state; (ii) the state variable is subject to a static constraint; (iii) the admissible controls are merely locally integrable in the right half-line. Such assumptions seem to be weaker than those appearing in most of the existing literature. We give a direct proof of the existence of an optimal control for any initial capital \(k_{0}\ge 0\) and we carry on a qualitative study of the value function; moreover, using dynamic programming methods, we show that the value function is a continuous viscosity solution of the associated Hamilton–Jacobi–Bellman equation.

Network Utility Maximization Under Maximum Delay Constraints and Throughput Requirements

Abstract: We consider a multi-path routing problem of maximizing the aggregate user utility over a multi-hop network, subject to link capacity constraints, maximum end-to-end delay constraints, and user throughput requirements. A user’s utility is a concave function of the achieved throughput or the experienced maximum delay. The problem is important for supporting real-time multimedia traffic and is uniquely challenging due to the need of simultaneously considering maximum delay constraints and throughput requirements. In this paper, we first show that it is NP-complete either (i) to construct a feasible solution strictly meeting all constraints, or (ii) to obtain an optimal solution after relaxing either the maximum delay constraints or the throughput requirements. We then develop a polynomial-time approximation algorithm named PASS . The design of PASS leverages a novel understanding between non-convex maximum-delay-aware problems and their convex average-delay-aware counterparts, which can be of independent interest and suggests a new avenue for solving maximum-delay-aware network optimization problems. We prove that PASS always obtains approximate solutions (i.e., with theoretical performance guarantees), at the cost of violating both the maximum delay constraints and the throughput requirements by up to constant ratios. We also develop two variants of PASS , named PASS-M and PASS-T , to generate approximate solutions at the cost of violating either the maximum delay constraints or the throughput requirements by up to problem-dependent ratios. We evaluate our solutions using extensive simulations on Amazon EC2 datacenters supporting video-conferencing traffic. Compared to the existing algorithms and a conceivable baseline, our solutions obtain up to 100% improvement of utilities, by meeting the throughput requirements but relaxing the maximum delay constraints to the extent acceptable for practical video conferencing applications.

Network Revenue Management with Inventory-Sensitive Bid Prices and Customer Choice

Abstract: We develop a new approximate dynamic programming approach to network revenue management models with customer choice that approximates the value function of the Markov decision process with a concave function which is separable across resource inventory levels. This approach reflects the intuitive interpretation of diminishing marginal utility of inventory levels and allows for significantly improved accuracy compared to currently available methods. The model allows for arbitrary aggregation of inventory units and thereby reduction of computational workload, yields upper bounds on the optimal expected revenue that are provably at least as tight as those obtained from previous approaches, and is asymptotically optimal under fluid scaling. Computational experiments for the multinomial logit choice model with distinct consideration sets show that policies derived from our approach outperform available alternatives, and we demonstrate how aggregation can be used to balance solution quality and runtime.

Optimum general threshold secret sharing

Abstract: An important issue of threshold secret sharing (TSS) schemes is to minimize the size of shares. This issue is resolved for the simpler classes called (k,n)-TSS and (k,L,n)-threshold ramp secret sharing (TRSS). That is, for each of these two classes, an optimum construction which minimizes the share size was presented. The goal of this paper is to develop an optimum construction for a more general threshold class where the mutual information between the secret and a set of shares is defined by a discrete function which monotonically increases from zero to one with the number of shares. A tight lower bound of the entropy of shares is first derived and then an optimum construction is presented. The derived lower bound is larger than the previous one except for special functions such as convex and concave functions. The optimum construction encodes the secret by using one or more optimum TRSS schemes independently. The optimality is shown by devising a combination of TRSS schemes which achieves the new lower bound.

Construction of interlaced polynomial lattice rules for infinitely differentiable functions

Abstract: We study multivariate integration over the s-dimensional unit cube in a weighted space of infinitely differentiable functions. It is known from a recent result by Suzuki that there exists a good quasi-Monte Carlo (QMC) rule which achieves a super-polynomial convergence of the worst-case error in this function space, and moreover, that this convergence behavior is independent of the dimension under a certain condition on the weights. In this paper we provide a constructive approach to finding a good QMC rule achieving such a dimension-independent super-polynomial convergence of the worst-case error. Specifically, we prove that interlaced polynomial lattice rules, with an interlacing factor chosen properly depending on the number of points N and the weights, can be constructed using a fast component-by-component algorithm in at most $$O(sN(\log N)^2)$$ arithmetic operations to achieve a dimension-independent super-polynomial convergence. The key idea for the proof of the worst-case error bound is to use a variant of Jensen’s inequality with a purposely-designed concave function.