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2 edition of Lipschitz continuous policy functions for strongly concave optimization problems found in the catalog.

Lipschitz continuous policy functions for strongly concave optimization problems

Luigi Montrucchio

# Lipschitz continuous policy functions for strongly concave optimization problems

## by Luigi Montrucchio

Published by Institute for Mathematical Studies in the Social Sciences, Stanford University in Stanford, Calif .
Written in English

Subjects:
• Social sciences -- Mathematical models.

• Edition Notes

The Physical Object ID Numbers Statement by Luigi Montrucchio. Series Technical report / Institute for Mathematical Studies in the Social Sciences, Stanford University -- no. 507, Economics series / Institute for Mathematical Studies in the Social Sciences, Stanford University, Technical report (Stanford University. Institute for Mathematical Studies in the Social Sciences) -- no. 507., Economics series (Stanford University. Institute for Mathematical Studies in the Social Sciences) Pagination 21 p. ; Number of Pages 21 Open Library OL22410239M

This paper shows that if the consumption good production function isα-concave and if the capital goods technologies are Lipschitz-continuous then the indirect utility function is strongly concave. Journal of Economic Literature Classification Numbers: C61, O iis L-Lipschitz continuous. If f iis -strongly convex-concave and its gradient is L-Lipschitz continuous, we deﬁne = L= be the condition number of f. In addition, the strongly convex-concave property in Assumption 1 means the gradient operator g iis monotone and the proximal operator is non-expansive. Lemma 1 (monotonicity [23]).Cited by: 2.

(a) Show that the composition of Lipschitz continuous functions is again Lipschitz continuous. (b) Is the pointwise maximum of two Lipschitz continuous functions necessarily Lipschitz continuous.   Lipschitz functions and convexity. Aug in Convex Optimization, (Lipschitz Continuous functions:) A function is said to be Lipschitz if. The advantage of a strongly convex function is that the function will not be flat at the bottom and hence the convergence to the optimal solution will be faster.

Download PDF: Sorry, we are unable to provide the full text but you may find it at the following location(s): (external link). Perturbation Analysis of Optimization Problems Iterative Algorithm for a System of Equilibrium Problems of Bregman Strongly Nonexpansive Mapping norm with l 2 -norm of Lipschitz continuous.

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### Lipschitz continuous policy functions for strongly concave optimization problems by Luigi Montrucchio Download PDF EPUB FB2

Montrucchio, Lipschitz continuous policy functions Under AA.4, problem P (xo, 8) has one and only one optimal solution (x*) for any given initial condition xo in X. Moreover, the value function' Wturns out to be strictly concave on X and to satisfy the Cited by: Montrucchio, Luigi, "Lipschitz continuous policy functions for strongly concave optimization problems," Journal of Mathematical Economics, Elsevier, vol.

16(3. In this chapter, we discuss global optimization problems where the functions involved are Lipschitz-continuous on certain subsets M ⊂ ℝ n. Section 1 presents a brief introduction into the most often treated univariate case. Section 2 is devoted to branch and bound : Reiner Horst, Hoang Tuy.

Lipschitz optimization solves global optimization problems in which the objective function and constraint left-hand sides may be given by oracles (or explicitly) and have a bounded slope. The problems of finding an optimal solution, an ε -optimal one, all optimal solutions, and a small volume enclosure of all optimal solutions within.

This paper contains basic results that are useful for building algorithms for the optimization of Lipschitz continuous functionsf on compact subsets of E n. In this setting f is differentiable a.e. The theory involves a set-valued mapping x→δ ∈ f(x) whose range is the convex hull of existing values of ∇ f and limits of ∇ f on a closed ∈ -ball, B(x, ∈).Cited by: Pinter's book can be recommended to each reader who needs a far-reaching survey of continuous and Lipschitz optimization.

The book is distinguished by its vivid and informative style, the large number of areas that are touched, the lack of longwinded and overly sophisticated expositions, and the optimum balance between theory and practical. We prove that the policy function, obtained by optimizing a discounted infinite sum of stationary return functions, is Lipschitz continuous when the instantaneous function is strongly concave.

Global optimization of Lipschitz functions The analysis provided in the paper considers that the number nof evaluation points is not ﬁxed and it is assumed that function evaluations are noiseless.

Moreover, the assumption made on the unknown function f throughout the paper is that it has a ﬁnite Lipschitz constant k, Size: KB. convex optimization problems. This makes it possible today to solve opti-mization problems that were previously out of reach.

The embryo of this book is a compendium written by Christer Borell and myself {79, but various additions, deletions and revisions over the years, have led to a completely di erent text.

Provide details and share your research. But avoid Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. Use MathJax to format equations.

Local strong convexity and local Lipschitz continuity of the gradient of convex functions R. Goebel∗ and R.T. Rockafellar† Abstract. Given a pair of convex conjugate functions f and f∗, we investigate the relationship between local Lipschitz continuity of ∇f and local strong convexity prop-erties of f∗.

Size: KB. Downloadable (with restrictions). This paper proves the C1,1 differentiability of the value function for continuous time concave dynamic optimization problems, under the assumption that the instantaneous utility is C1,1 and the initial segment of optimal solutions is interior. From this result, the Lipschitz dependence of optimal solutions on initial data and the Lipschitz continuity of the.

First, some definitions: Lipschitz continuity. A function is called L-Lipschitz over a set S with respect to a norm if for all we have. Some people will equivalently say is Lipschitz continuous with Lipschitz constant. Intuitively, is a measure of how fast the function can change. Convexity, subgradients.

function is convex if. Convex, concave, strictly convex, and strongly convex functions First and second order characterizations of convex functions Optimality conditions for convex problems 1 Theory of convex functions De nition Let’s rst recall the de nition of a convex function.

De nition 1. A function f: Rn!Ris convex if its domain is a convex set and for File Size: 1MB. In this paper we propose a new simplicial partition-based deterministic algorithm for global optimization of Lipschitz-continuous functions without requiring any knowledge of the Lipschitz constant.

Our algorithm is motivated by the well-known Direct algorithm which evaluates the objective function on a set of points that tries to cover the most promising subregions of the feasible region. Abstract. We study generalized parametric optimization problems in Banach spaces, given by continuously Fréchet differentiable mappings and some abstract constraints, in terms of local Lipschitz continuity of the optimal value function.

Lipschitz continuous. Polyak [] This inequality simply requires that the gradient grows faster than a linear function as we move away from the optimal function value. Invex function (one global minimum) Invex functions (a generalization of convex function) Assumptions Objective function.

Lipschitz continuous. Download PDF: Sorry, we are unable to provide the full text but you may find it at the following location(s): (external link)Author: Luigi Montrucchio. STRONGLY CONCAVE PROBLEMS As a first application of theorem 1 let us prove that the optimal policy function of an optimal growth model (SZ, V) with a strongly concave utility function Y is Holder continuous in the interior of the state by: 8.

An α-concave function is also called a strongly concave function, with modulus α. Analogous expressions are used for α x and α y concave functions. The following hypotheses will. Global optimization of Lipschitz functions 2.

Setup and preliminary results Setup and notations Setup. Let XˆRdbe a compact and convex set with non-empty interior and let f: X!R be an unknown function which is only supposed to admit a maximum over its in-put domain X. The goal in global optimization consists in ﬁnding some point x Cited by: The above definition can be used to show that the Hessian of a strongly Convex function must satisfy.

Convex optimization problems are an important class (subsumes linear and contains a subset of non-linear problems) that are interesting, useful and that can be solved efficiently. A function is Lipschitz continuous with constant if.It is shown that solutions of linear inequalities, linear programs and certain linear complementarity problems (e.g.

those with P-matrices or Z-matrices but not semidefinite matrices) are Lipschitz continuous with respect to changes in the right-hand side data of the ons of linear programs are not Lipschitz continuous with respect to the coefficients of the objective by: