Showing papers on "Concave function published in 1973"
TL;DR: This paper has shown that any maximizing sequence for the dual can be made to yield, in a general way, an asymptotically minimizingsequence for the primal which typically converges at least as rapidly.
Abstract: Several recent algorithms for solving nonlinear programming problems with equality constraints have made use of an augmented “penalty” Lagrangian function, where terms involving squares of the constraint functions are added to the ordinary Lagrangian. In this paper, the corresponding penalty Lagrangian for problems with inequality constraints is described, and its relationship with the theory of duality is examined. In the convex case, the modified dual problem consists of maximizing a differentiable concave function (indirectly defined) subject to no constraints at all. It is shown that any maximizing sequence for the dual can be made to yield, in a general way, an asymptotically minimizing sequence for the primal which typically converges at least as rapidly.
387 citations
TL;DR: It is shown that the rate-distortion bound (R(d) \leq C) remains true when -\log x in the definition of mutual information is replaced by an arbitrary concave nonincreasing function satisfying some technical conditions.
Abstract: It is shown that the rate-distortion bound (R(d) \leq C) remains true when -\log x in the definition of mutual information is replaced by an arbitrary concave (\cup) nonincreasing function satisfying some technical conditions. Examples are given showing that for certain choices of the concave functions, the bounds obtained are better than the classical rate-distortion bounds.
93 citations
TL;DR: This paper gives counterexamples to: 1 Ritter's algorithm for the global maximization of a quadratic subject to linear inequality constraints, and 2 Tui's algorithmfor the global minimizations of a concave function subject tolinear inequality constraints.
Abstract: This paper gives counterexamples to: 1 Ritter's algorithm for the global maximization of a quadratic subject to linear inequality constraints, and 2 Tui's algorithm for the global minimization of a concave function subject to linear inequality constraints.
67 citations
TL;DR: In this paper, a branch and bound technique is used to identify the global minimum extreme point of a convex polyhedron and a linear undrestimator for the constrained concave objective function is developed.
Abstract: A general algorithm is developed for minimizing a well defined concave function over a convex polyhedron. The algorithm is basically a branch and bound technique which utilizes a special cutting plane procedure to' identify the global minimum extreme point of the convex polyhedron. The indicated cutting plane method is based on Glover's general theory for constructing legitimate cuts to identify certain points in a given convex polyhedron. It is shown that the crux of the algorithm is the development of a linear undrestimator for the constrained concave objective function. Applications of the algorithm to the fixed-charge problem, the separable concave programming problem, the quadratic problem, and the 0-1 mixed integer problem are discussed. Computer results for the fixed-charge problem are also presented.
46 citations
01 Jan 1973
TL;DR: In this article, the attainable set V and Pareto surface P are defined for concave continuous functions defined over the unit m-cube Im, and the notion of complexity (the smallest m for which a given V can be realized) is briefly discussed.
Abstract: Given n concave continuous functions u, defined over the unit m-cube Im, the corresponding attainable set V and Pareto surface P are defined. In the economic interpretation, V corresponds to the set of attainable utility outcomes realized through trading, and P the set of such outcomes for which no trader can attain more without another getting less. Sets of the form of V and P are characterized among all subsets of Rn. The notion of complexity (the smallest m for which a given V can be realized) is briefly discussed, as is the idea of a "market game".
38 citations
TL;DR: It is shown that the class of functions that are both " generalized concave" and "generalized convex" is closely related to the classOf monotone functions of a single variable, which means that, for arbitrary perhaps not differentiable functions, concave implies pseudoconcave, pseudconcave implies strictly quasiconc Cave, and strictly quAsiconcave imply quasIconcave.
Abstract: This paper examines properties and interrelations of several concepts of generalized concavity It shows that the class of functions that are both "generalized concave" and "generalized convex" is closely related to the class of monotone functions of a single variable After excluding a certain small class of exceptions, the paper shows that, for arbitrary perhaps not differentiable functions, concave implies pseudoconcave, pseudoconcave implies strictly quasiconcave, and strictly quasiconcave implies quasiconcave Several results characterizing the extreme values of generalized concave functions are given
35 citations
TL;DR: The main purpose of as discussed by the authors is to characterize those functions f : B → R + such that (1) holds, i.e., if ξ is chosen such that 1 ¦B¦ ∝ B ϑ(f(x)) dx = n ∝ 0 1 ϑ (ξt)(1 − t) n − 1) dt.
23 citations
TL;DR: In this paper, a multidimensional inverse Minkowski inequality for concave functions was shown for functions f 1 a 1, 1, 2, f n a n (f k concave, 1 dimension), where ∂ 2 f k ∂x i 2 ⩽ 0, i = 1,…, d, k = 1.
19 citations
8 citations
01 May 1973
TL;DR: Many equilibrium and optimization problems in economics and operations research can be put in the form: Find x such that f(x) = x, where x is a nonnegative vector with component sum one and f is a continuous function.
Abstract: Many equilibrium and optimization problems in economics and operations research can be put in the form: Find x such that f(x) = x, where x is a nonnegative vector with component sum one and f is a continuous function (not necessarily differentiable, convex, or concave).
4 citations
TL;DR: In this paper, the authors consider the problem of determining if a local minimization of a real-valued function in a closed convex subset of a Hilbert space lies nearby and if so, to estimate its distance.
Abstract: If C is a closed convex subset of a Hilbert space H, and f a real-valued function defined and twice Frechet differentiable on C, we consider whether, given a point $x_0 \in H$, it is possible to determine if a local minimizes of f in C lies nearby, and if so, to estimate its distance.
TL;DR: In this article, a special min-max problem is reduced to a certain type of problem in which a concave function is to be minimized over a convex set, and a solution procedure for this problem is developed and the first feasible solution yields a t the same time the global minimum.
Abstract: In this paper special min-max problems are reduced to a certain type of problem in which a concave function is to be minimized over a convex set. On the basis of the extension principle a solution procedure for this problem is developed and the first feasible solution yields a t the same time the global minimum. Since the feasible set is defined by linear constraints, the algorithm consists essentially in the application of the simplex transformation.