Topic
Concave function
About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.
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TL;DR: This paper considers a problem with general concave costs at the supply points, as well as convex Costs at the demand points, and proposes a solution method which reduces the problem to a d.c. function, and proves convergence of the method and report computational tests that indicate that quite large problems can be solved efficiently.
Abstract: The problem of transporting goods from a set of supply points (factories) to a set of demand points (customers) so as to minimize linear transportation costs is well known and very efficient solution methods exist. In the well-known facility location problem one also includes fixed charges for the supply points. On the other hand, in ordinary transportation problems, stochastic demand has been introduced and modelled by convex costs, yielding the Stochastic Transportation Problem, [STP], solved by for example the methods in Cooper and LeBlanc (1977) and Holmberg and Jornsten (1984). However, the simultaneous use of these generalizations has recieved little attention until now. Only a few suggestions for solution methods can be found, LeBlanc (1977), Franca and Luna (1982). In this paper we consider generalizing this problem even further, by introducing general concave costs at the supply points, as well as convex costs at the demand points. Fixed charges are not the only cost structures that occur when producing goods at a factory. Economies of scale very often yield other concave cost functions.
82 citations
TL;DR: The heavy-quark--antiquark potential is shown to be a monotone nondecreasing and concave function of the separation.
Abstract: The heavy-quark--antiquark potential is shown to be a monotone nondecreasing and concave function of the separation. This property holds independent of the gauge group and the details of the matter sector.
82 citations
12 Dec 2005
TL;DR: This work provides a centralized algorithm for off-line analysis and establishment of a performance benchmark for nonconcave utility maximization, and employs convex sum-of-squares relaxations solved by a sequence of semidefinite programs to obtain increasingly tighter upper bounds on total achievable utility for polynomial utilities.
Abstract: The Network Utility Maximization problem has recently been used extensively to analyze and design distributed rate allocation in networks such as the Internet. A major limitation in the state-of-the-art is that user utility functions are assumed to be strictly concave functions, modeling elastic flows. Many applications require inelastic flow models where nonconcave utility functions need to be maximized. It has been an open problem to find the globally optimal rate allocation that solves nonconcave network utility maximization, which is a difficult nonconvex optimization problem. We provide a centralized algorithm for off-line analysis and establishment of a performance benchmark for nonconcave utility maximization. Based on the semialgebraic approach to polynomial optimization, we employ convex sum-of-squares relaxations solved by a sequence of semidefinite programs, to obtain increasingly tighter upper bounds on total achievable utility for polynomial utilities. Surprisingly, in all our experiments, a very low order and often a minimal order relaxation yields not just a bound on attainable network utility, but the globally maximized network utility. When the bound is exact, which can be proved using a sufficient test, we can also recover a globally optimal rate allocation. In addition to polynomial utilities, sigmoidal utilities can be transformed into polynomials and are handled. Furthermore, using two alternative representation theorems for positive polynomials, we present price interpretations in economics terms for these relaxations, extending the classical interpretation of independent congestion pricing on each link to pricing for the simultaneous usage of multiple links.
81 citations
TL;DR: Simulations show that the improved model proposed can reproduce well the metastable state, the spatiotemporal patterns, and the phase transitions of traffic flow and can be simulated with a quantitative agreement.
Abstract: This paper firstly shows that a recent model (Tian et al., Transpn. Res. B 71, 138–157, 2015) is not able to replicate well the concave growth pattern of traffic oscillations (i.e., the standard deviation of speed is a concave function of the vehicle number in the platoon) observed from car following experiments. We propose an improved model by introducing a safe speed and the logistic function for the randomization probability. Simulations show that the improved model can reproduce well the metastable state, the spatiotemporal patterns, and the phase transitions of traffic flow. Calibration and validation results show that the concave growth pattern of oscillations and the empirical detector data can be simulated with a quantitative agreement.
81 citations
TL;DR: This paper is concerned with the problem of finding the global minimum of a concave function over a closed, convex, possibly unbounded set in R n.
Abstract: In this paper we are concerned with the problem of finding the global minimum of a concave function over a closed, convex, possibly unbounded set in Rn. The intrinsic difficulty of this problem is due to the fact that a local minimum of the objective function may fail to be a global one—which makes the conventional methods of local optimization almost useless.
81 citations