A function defined on a locally convex space is called evenly quasiconvex if its level sets are intersections of families of open half spaces. Furthermore, if the closures of these open halfspaces do not contain the o...
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A function defined on a locally convex space is called evenly quasiconvex if its level sets are intersections of families of open half spaces. Furthermore, if the closures of these open halfspaces do not contain the origin, then the function is called R-evenly quasiconvex. In this note, R-evenly quasiconvex functions are characterized as those evenly-quasiconvex functions that satisfy a certain simple relation with their lower semicontinuous hulls.
We introduce the notion of variational (semi-) strict quasimonotonicity for a multivalued operator T : X double right arrow X* relative to a nonempty subset A of X which is not necessarily included in the domain of T....
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We introduce the notion of variational (semi-) strict quasimonotonicity for a multivalued operator T : X double right arrow X* relative to a nonempty subset A of X which is not necessarily included in the domain of T. We use this notion to characterize the subdifferentials of continuous (semi-) strictly quasiconvex functions. The proposed definition is a relaxation of the standard definition of (semi-) strict quasimonotonicity, the latter being appropriate only for operators with nonempty values. Thus, the derived results are extensions to the continuous case of the corresponding results for locally Lipschitz functions.
We establish existence of steepest descent curves emanating from almost every point of a regular locally Lipschitz quasiconvex functions, where regularity means that the sweeping process flow induced by the sublevel s...
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We establish existence of steepest descent curves emanating from almost every point of a regular locally Lipschitz quasiconvex functions, where regularity means that the sweeping process flow induced by the sublevel sets is reversible. We then use max-convolution to regularize general quasiconvex functions and obtain a result of the same nature in a more general setting.
This paper studies the convergence of the proximal point method for quasiconvex functions in finite dimensional complete Riemannian manifolds. We prove initially that, in the general case, when the objective function ...
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This paper studies the convergence of the proximal point method for quasiconvex functions in finite dimensional complete Riemannian manifolds. We prove initially that, in the general case, when the objective function is proper and lower semicontinuous, each accumulation point of the sequence generated by the method, if it exists, is a limiting critical point of the function. Then, under the assumptions that the sectional curvature of the manifold is bounded above by some non negative constant and the objective function is quasiconvex we analyze two cases. When the constant is zero, the global convergence of the algorithm to a limiting critical point is assured and if it is positive, we prove the local convergence for a class of quasiconvex functions, which includes Lipschitz functions.
In this paper we show how to approximate a quasiconvex function with a sequence of strictly quasiconvex functions in a reflexive Banach space X. An important role in our approximation procedure is played by a real val...
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In this paper we show how to approximate a quasiconvex function with a sequence of strictly quasiconvex functions in a reflexive Banach space X. An important role in our approximation procedure is played by a real valued convex function defined on X, and parameterized by a pair of closed bounded convex sets, which is a generalization of the classical Minkowski functional on X;for this reason, we investigate some of its properties. In particular, we prove the continuity of this map, seen as a function acting from a specific family of pairs of closed convex subsets of X, to the space of the real valued continuous functions on X. In the domain space we use the (bounded) Hausdorff topology, while the target space is endowed with the topology of the uniform convergence on bounded sets. Our results also need to approximate a closed convex set, in the sense of the bounded Hausdorff topology, with a sequence of strictly convex sets. The result particularizes to Hausdorff topology if the limit set is bounded.
In this article we study a Hadamard type inequality for nonnegative evenly quasiconvex functions. The approach of our study is based on the notion of abstract convexity. We also provide an explicit calculation to eval...
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In this article we study a Hadamard type inequality for nonnegative evenly quasiconvex functions. The approach of our study is based on the notion of abstract convexity. We also provide an explicit calculation to evaluate the asymptotically sharp constant associated with the inequality over a unit square in the two-dimensional plane. (C) 2002 Elsevier Science (USA). All rights reserved.
This paper extends the full convergence of the steepest descent method with a generalized Armijo search and a proximal regularization to solve minimization problems with quasiconvex objective functions on complete Rie...
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This paper extends the full convergence of the steepest descent method with a generalized Armijo search and a proximal regularization to solve minimization problems with quasiconvex objective functions on complete Riemannian manifolds. Previous convergence results are obtained as particular cases and some examples in non-Euclidian spaces are given. In particular, our approach can be used to solve constrained minimization problems with nonconvex objective functions in Euclidian spaces if the set of constraints is a Riemannian manifold and the objective function is quasiconvex in this manifold. (C) 2007 Elsevier Inc. All rights reserved.
The article is concerned with an optimization problem where we minimize the difference of two quasiconvex scalar functions under a vector-valued quasiconvex system. Using the Q-subdifferential introduced by Suzuki and...
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The article is concerned with an optimization problem where we minimize the difference of two quasiconvex scalar functions under a vector-valued quasiconvex system. Using the Q-subdifferential introduced by Suzuki and Kuroiwa [Optimality conditions and the basic constraint qualification for quasiconvex programming. Nonlinear Anal. 2011;74:1279?1285], together with a special constraint qualification, we give necessary optimality conditions to (P).
We study the class Q of quasiconvex functions (i.e. functions with convex sublevel sets), by associating to every u is an element of Q boolean AND C(R-n) a function H:R-n x R -> R boolean OR {+/-infinity}, such tha...
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We study the class Q of quasiconvex functions (i.e. functions with convex sublevel sets), by associating to every u is an element of Q boolean AND C(R-n) a function H:R-n x R -> R boolean OR {+/-infinity}, such that H(X, t) is nondecreasing in t and sublinear in X: for every fixed t, the function H(center dot, t) is nothing else than the support function of the sublevel set {x is an element of R-n: u(x) <= t}. When u is suitably regular, we establish an exact relation between D(2)u and (DH)-H-2;this allows us to find explicit formulae to write the k-Hessian operators S-k(D(2)u) (among which Delta u and det D(2)u) in terms of H. Then we investigate on Minkowski addition of quasiconvex functions. (C) 2007 Elsevier Masson SAS. All rights reserved.
Letf be a real-valued function defined on the product ofm finite-dimensional open convex setsX1, ⋯,X m .Assume thatf is quasiconvex and is the sum of nonconstant functionsf1, ⋯,f m defined on the respective factor set...
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Letf be a real-valued function defined on the product ofm finite-dimensional open convex setsX1, ⋯,X m .Assume thatf is quasiconvex and is the sum of nonconstant functionsf1, ⋯,f m defined on the respective factor sets. Then everyf i is continuous; with at most one exception every functionf i is convex; if the exception arises, all the other functions have a strict convexity property and the nonconvex function has several of the differentiability properties of a convex *** define the convexity index of a functionf i appearing as a term in an additive decomposition of a quasiconvex function, and we study the properties of that index. In particular, in the case of two one-dimensional factor sets, we characterize the quasiconvexity of an additively decomposed functionf either in terms of the nonnegativity of the sum of the convexity indices off1 andf2, or, equivalently, in terms of the separation of the graphs off1 andf2 by means of a logarithmic function. We investigate the extension of these results to the case ofm factor sets of arbitrary finite dimensions. The introduction discusses applications to economic theory.
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