Concave function

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

Characterizing properties of stochastic objective functions

Abstract: This paper develops tools for analyzing properties of stochastic objective functions that take the form ( , ) ( , ) ( ; ) Vu dF ≡ ∫s xx s s θ θ . The paper analyzes the relationship between properties of the primitive functions, the utility function u and probability distribution F, and properties of the stochastic objective. The methods apply when the utility function is restricted to lie in a set of functions which is a "closed convex cone" (e.g., nondecreasing functions, concave functions, or supermodular functions). Approaches previously applied to characterize monotonicity of V (that is, stochastic dominance theorems) can be used to establish other properties of V as well. The first part of the paper establishes necessary and sufficient conditions for V to satisfy "closed convex cone properties," such as supermodularity, in the parameter θ. Then, we consider necessary and sufficient conditions for monotone comparative statics predictions. A new property of payoff functions is introduced, called l-supermodularity, which is shown to be necessary and sufficient for comparative statics predictions. The results are illustrated with applications.

Relaxed performance measure approach for reliability-based design optimization

Abstract: The efficiency and robustness of reliability analysis methods are important factors to evaluate the probabilistic constraints in reliability-based design optimization (RBDO). In this paper, a relaxed mean value (RMV) approach is proposed in order to evaluate probabilistic constraints including convex and concave functions in RBDO using the performance measure approach (PMA). A relaxed factor is adaptively determined in the range from 0 to 2 using an inequality criterion to improve the efficiency and robustness of the inverse first-order reliability methods. The performance of the proposed RMV is compared with six existing reliability methods, including the advanced mean value (AMV), conjugate mean value (CMV), hybrid mean value (HMV), chaos control (CC), modified chaos control (MCC), and conjugate gradient analysis (CGA) methods, through four nonlinear concave and convex performance functions and three RBDO problems. The results demonstrate that the proposed RMV is more robust than the AMV, CMV, and HMV for highly concave problems, and slightly more efficient than the CC, MCC, and CGA methods. Furthermore, the proposed relaxed mean value guarantees robust and efficient convergence for RBDO problems with highly nonlinear performance functions.

Minimization of a quasi-concave function over an efficient set

Abstract: The nonconvex programming problem of minimizing a quasi-concave function over an efficient (or weakly efficient) set of a multiobjective linear program is studied. A cutting plane algorithm which finds an approximate optimal solution in a finite number of steps is developed. For the particular “all linear” case the algorithm performs better, finding an optimal solution in a finite time, and being more easily implemented.

On the global minimization of concave functions

Abstract: Many important classes of decision models give rise to the problem of finding a global minimum of a concave function over a convex set. Since many local minima can occur, concave minimization belongs to the "hard" global optimization problems, where standard nonlinear programming procedures fail. After a brief survey on important classes of decision models that can be formulated as concave minimization problems, the two main solution approaches are discussed: branch-and-bound combined with convex underestimation and outer approximation by cutting planes. Eine Reihe wichtiger Klassen von Entscheidungsmodellen fuhrt auf die globale Minimierung konkaver Funktionen uber konvexen Mengen. Da viele lokale Minima auftreten konnen, ist die konkave Minimierung zu den "harten" globalen Optimierungsaufgaben zu rechnen, bei denen Standardverfahren der nichtlinearen Optimierung nicht zum Ziel fuhren. Nach einer kurzen Ubersicht uber wichtige Klassen von Entscheidungsmodellen, die sich als konkave Minimierungsprobleme formulieren lassen, werden die beiden wichtigsten Losungsansatze diskutiert: Eine Kombination von Branch und Bound mit konvexer Approximation und auβere Approximationsverfahren mit Hilfe von Schnittebenen.