Concave function

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

Stochastic Scheduling on Parallel Processors and Minimization of Concave Functions of Completion Times

Abstract: We consider a stochastic scheduling problem in which n jobs are to be scheduled on m identical processors which operate in parallel. The processing times of the jobs are not known in advance but they have known distributions with hazard rates ρ 1, (t), …, ρ n (t). It is desired to minimize the expected value of к(C), where C i is the time at which job i is completed C = (C 1, …, C n ), and к(C) is increasing and concave in C. Suppose processor i first becomes available at time τ i . We prove that if there is a single static list priority policy which is optimal for every τ = (τ 1, …, τ m ), then the minimal expected cost must be increasing and concave in τ. Moreover, if к(C) is supermodular in C then this cost is also supermodular in τ. This result is used to prove that processing jobs according to the static list priority order (1,2,…,n) minimizes the expected value of ∑w i h(C i ), when h(·) is a nondecreasing, concave function, w 1 ≥ … ≥ w n , and ρ 1 (t 1)w 1 ≥ … ≥ ρ n (t n )w n for all t 1, …, t n .

Automatic continuity of concave functions

Abstract: A necessary and sufficient condition is given that a semicontinuous, nonnegative, concave function on a finite dimensional closed convex set X necessarily be continuous at a point xo E X. Application of this criterion at all points of X yields a characterization, due to Gale, Klee and Rockafellar, of convex polyhedra in terms of continuity of their convex functions. Let V be a real vector space of dimension n 0, being closed. Observe also that the L+(0, s) are convex. We say X is polyhedral if it is specified by a finite number of linear inequalities (3) X = {v E V: Ai(v) ? bi, Ai E V*, bi E R, 1 0 we have a closed convex set of Y C V x [0, t] such that (4) (a) Yn(V x {0}) =Xx {0} (b) If (x, r) E Y, then (x, r') E Y for 0 < r' < r. Received by the editors December 18, 1986 and, in revised form, May 4, 1987. 1980 Mathernatic8s Subject (lassification (1985 R?evision). Primary 52A20.

A unified DC programming framework and efficient DCA based approaches for large scale batch reinforcement learning

Abstract: We investigate a powerful nonconvex optimization approach based on Difference of Convex functions (DC) programming and DC Algorithm (DCA) for reinforcement learning, a general class of machine learning techniques which aims to estimate the optimal learning policy in a dynamic environment typically formulated as a Markov decision process (with an incomplete model). The problem is tackled as finding the zero of the so-called optimal Bellman residual via the linear value-function approximation for which two optimization models are proposed: minimizing the $$\ell _{p}$$ -norm of a vector-valued convex function, and minimizing a concave function under linear constraints. They are all formulated as DC programs for which attractive DCA schemes are developed. Numerical experiments on various examples of the two benchmarks of Markov decision process problems—Garnet and Gridworld problems, show the efficiency of our approaches in comparison with two existing DCA based algorithms and two state-of-the-art reinforcement learning algorithms.

Minimax Theorems for Extended Real-Valued Abstract Convex–Concave Functions

Abstract: In this paper, we provide sufficient and necessary conditions for the minimax equality for extended real-valued abstract convex–concave functions. As an application, we get sufficient and necessary conditions for the minimax equality for extended real-valued convex–concave functions.