Techniques of Variational Analysis
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Applications of Variational Analysis to a Generalized Heron Problem Boris S
Wayne State University Mathematics Research Reports Mathematics 7-1-2011 Applications of Variational Analysis to a Generalized Heron Problem Boris S. Mordukhovich Wayne State University, [email protected] Nguyen Mau Nam The University of Texas-Pan American, Edinburg, TX, [email protected] Juan Salinas Jr The University of Texas-Pan American, Edinburg, TX, [email protected] Recommended Citation Mordukhovich, Boris S.; Nam, Nguyen Mau; and Salinas, Juan Jr, "Applications of Variational Analysis to a Generalized Heron Problem" (2011). Mathematics Research Reports. Paper 88. http://digitalcommons.wayne.edu/math_reports/88 This Technical Report is brought to you for free and open access by the Mathematics at DigitalCommons@WayneState. It has been accepted for inclusion in Mathematics Research Reports by an authorized administrator of DigitalCommons@WayneState. APPLICATIONS OF VARIATIONAL ANALYSIS TO A GENERALIZED HERON PROBLEM BORIS S. MORDUKHOVICH, NGUYEN MAU NAM and JUAN SALINAS JR. WAYNE STATE UNIVERSilY Detroit, Ml 48202 Department of Mathematics Research Report 2011 Series #7 . APPLICATIONS OF VARIATIONAL ANALYSIS TO A GENERALIZED HERON PROBLEM 2 3 BORIS S. MORDUKHOVICH1 , NGUYEN MAU NAM and JUAN SALINAS JR. Abstract. This paper is a continuation of our ongoing efforts to solve a number of geometric problems and their extensions by using advanced tools of variational analysis and generalized differentiation. Here we propose and study, from both qualitative and numerical viewpoints, the following optimal location problem as well as its further extensions: on a given nonempty subset of a Banach space, find a point such that the sum of the distances from it to n given nonempty subsets of this space is minimal. -
Variational Analysis in Normed Spaces with Applications to Constrained
Variational Analysis in Normed Spaces with Applications to Constrained Optimization∗ ASHKAN MOHAMMADI† and BORIS S. MORDUKHOVICH‡ Abstract. This paper is devoted to developing and applications of a generalized differential theory of varia- tional analysis that allows us to work in incomplete normed spaces, without employing conventional variational techniques based on completeness and limiting procedures. The main attention is paid to generalized derivatives and subdifferentials of the Dini-Hadamard type with the usage of mild qualification conditions revolved around metric subregularity. In this way we develop calculus rules of generalized differentiation in normed spaces without imposing restrictive normal compactness assumptions and the like and then apply them to general problems of constrained optimization. Most of the obtained results are new even in finite dimensions. Finally, we derive refined necessary optimality conditions for nonconvex problems of semi-infinite and semidefinite programming. Keywords. Variational analysis, constrained optimization, generalized differentiation, metric subregularity, semi- infinite programming, semidefinite programming. Mathematics Subject Classification (2000) 49J52, 49J53, 90C48, 90C34 1 Introduction A characteristic feature of modern variational analysis in both finite and infinite dimensions is a systematic usage of perturbation, approximation, and hence limiting procedures; see, e.g., the books [3,5,10,18,24,25, 27,30] and the references therein. In infinite dimensions these procedures require the completeness of the space in question that plays a crucial role in the development and applications of major variational and extremal principles of variational analysis. It has been well recognized that such principles (in particular, the most powerful Ekeland variational principle) are at the heart of variational techniques to derive calculus rules of generalized differentiation with their applications to optimization and control problems. -
Ekeland Variational Principles with Set-Valued Objective Functions And
Ekeland variational principles with set-valued objective functions and set-valued perturbations1 QIU JingHui School of Mathematical Sciences, Soochow University, Suzhou 215006, China Email: [email protected] Abstract In the setting of real vector spaces, we establish a general set-valued Ekeland variational principle (briefly, denoted by EVP), where the objective func- tion is a set-valued map taking values in a real vector space quasi-ordered by a convex cone K and the perturbation consists of a K-convex subset H of the or- dering cone K multiplied by the distance function. Here, the assumption on lower boundedness of the objective function is taken to be the weakest kind. From the general set-valued EVP, we deduce a number of particular versions of set-valued EVP, which extend and improve the related results in the literature. In particu- lar, we give several EVPs for approximately efficient solutions in set-valued opti- mization, where a usual assumption for K-boundedness (by scalarization) of the objective function’s range is removed. Moreover, still under the weakest lower boundedness condition, we present a set-valued EVP, where the objective function is a set-valued map taking values in a quasi-ordered topological vector space and the perturbation consists of a σ-convex subset of the ordering cone multiplied by the distance function. arXiv:1708.05126v1 [math.FA] 17 Aug 2017 Keywords nonconvex separation functional, vector closure, Ekeland variational principle, coradiant set, (C, ǫ)-efficient solution, σ-convex set MSC(2010) 46A03, 49J53, 58E30, 65K10, 90C48 1 Introduction Since the variational principle of Ekeland [12, 13] for approximate solutions of nonconvex minimization problems appeared in 1972, there have been various 1This work was supported by the National Natural Science Foundation of China (Grant Nos. -
Subderivative-Subdifferential Duality Formula
Subderivative-subdifferential duality formula Marc Lassonde Universit´edes Antilles, BP 150, 97159 Pointe `aPitre, France; and LIMOS, Universit´eBlaise Pascal, 63000 Clermont-Ferrand, France E-mail: [email protected] Abstract. We provide a formula linking the radial subderivative to other subderivatives and subdifferentials for arbitrary extended real-valued lower semicontinuous functions. Keywords: lower semicontinuity, radial subderivative, Dini subderivative, subdifferential. 2010 Mathematics Subject Classification: 49J52, 49K27, 26D10, 26B25. 1 Introduction Tyrrell Rockafellar and Roger Wets [13, p. 298] discussing the duality between subderivatives and subdifferentials write In the presence of regularity, the subgradients and subderivatives of a function f are completely dual to each other. [. ] For functions f that aren’t subdifferentially regular, subderivatives and subgradients can have distinct and independent roles, and some of the duality must be relinquished. Jean-Paul Penot [12, p. 263], in the introduction to the chapter dealing with elementary and viscosity subdifferentials, writes In the present framework, in contrast to the convex objects, the passages from directional derivatives (and tangent cones) to subdifferentials (and normal cones, respectively) are one-way routes, because the first notions are nonconvex, while a dual object exhibits convexity properties. In the chapter concerning Clarke subdifferentials [12, p. 357], he notes In fact, in this theory, a complete primal-dual picture is available: besides a normal cone concept, one has a notion of tangent cone to a set, and besides a subdifferential for a function one has a notion of directional derivative. Moreover, inherent convexity properties ensure a full duality between these notions. [. ]. These facts represent great arXiv:1611.04045v2 [math.OC] 5 Mar 2017 theoretical and practical advantages. -
Rational Approximation of the Absolute Value Function from Measurements: a Numerical Study of Recent Methods
Rational approximation of the absolute value function from measurements: a numerical study of recent methods I.V. Gosea∗ and A.C. Antoulasy May 7, 2020 Abstract In this work, we propose an extensive numerical study on approximating the absolute value function. The methods presented in this paper compute approximants in the form of rational functions and have been proposed relatively recently, e.g., the Loewner framework and the AAA algorithm. Moreover, these methods are based on data, i.e., measurements of the original function (hence data-driven nature). We compare numerical results for these two methods with those of a recently-proposed best rational approximation method (the minimax algorithm) and with various classical bounds from the previous century. Finally, we propose an iterative extension of the Loewner framework that can be used to increase the approximation quality of the rational approximant. 1 Introduction Approximation theory is a well-established field of mathematics that splits in a wide variety of subfields and has a multitude of applications in many areas of applied sciences. Some examples of the latter include computational fluid dynamics, solution and control of partial differential equations, data compression, electronic structure calculation, systems and control theory (model order reduction reduction of dynamical systems). In this work we are concerned with rational approximation, i.e, approximation of given functions by means of rational functions. More precisely, the original to be approximated function is the absolute value function (known also as the modulus function). Moreover, the methods under consideration do not necessarily require access to the exact closed-form of the original function, but only to evaluations of it on a particular domain. -
Best L1 Approximation of Heaviside-Type Functions in a Weak-Chebyshev Space
Best L1 approximation of Heaviside-type functions in Chebyshev and weak-Chebyshev spaces Laurent Gajny,1 Olivier Gibaru1,,2 Eric Nyiri1, Shu-Cherng Fang3 Abstract In this article, we study the problem of best L1 approximation of Heaviside-type functions in Chebyshev and weak-Chebyshev spaces. We extend the Hobby-Rice theorem [2] into an appro- priate framework and prove the unicity of best L1 approximation of Heaviside-type functions in an even-dimensional Chebyshev space under the condition that the dimension of the subspace composed of the even functions is half the dimension of the whole space. We also apply the results to compute best L1 approximations of Heaviside-type functions by polynomials and Her- mite polynomial splines with fixed knots. Keywords : Best approximation, L1 norm, Heaviside function, polynomials, polynomial splines, Chebyshev space, weak-Chebyshev space. Introduction Let [a, b] be a real interval with a < b and ν be a positive measure defined on a σ-field of subsets of [a, b]. The space L1([a, b],ν) of ν-integrable functions with the so-called L1 norm : b k · k1 : f ∈ L1([a, b],ν) 7−→ kfk1 = |f(x)| dν(x) Za forms a Banach space. Let f ∈ L1([a, b],ν) and Y be a subset of L1([a, b],ν) such that f∈ / Y where Y is the closure of Y . The problem of the best L1 approximation of the function f in Y consists in finding a function g∗ ∈ Y such that : ∗ arXiv:1407.0228v3 [math.FA] 26 Oct 2015 kf − g k1 ≤ kf − gk1, ∀g ∈ Y. -
Chapter 1 COMPACTNESS,OPTIMALITY and RISK
Chapter 1 COMPACTNESS,OPTIMALITY AND RISK B. Cascales, J. Orihuela and M. Ruiz Galan´ Summary: This is a survey about one of the most important achievements in op- timization in Banach space theory, namely, James’ weak compactness theorem, its relatives and its applications. We present here a good number of topics related to James’ weak compactness theorem and try to keep the technicalities needed as sim- ple as possible: Simons’ inequality is our preferred tool. Besides the expected ap- plications to measures of weak noncompactess, compactness with respect to bound- aries, size of sets of norm-attaining functionals, etc., we also exhibit other very recent developments in the area. In particular we deal with functions and their level sets to study a new Simons’ inequality on unbounded sets that appear as the epi- graph of some fixed function f . Applications to variational problems for f and to risk measures associated with its Fenchel conjugate f ∗ are studied. Key words: compactness, optimization, risk measures, measure of non weak- compactness, Simons’ inequality, nonattaining functionals, I-generation, variational problems, reflexivity. AMS 2010 Subject Classification: 46B10, 46B26, 46B50, 46E30, 46N10, 47J20, 49J52, 91B30, 91G10, 91G80. Dedicated to Jonathan Borwein on the occasion of his 60th Birthday B. Cascales, Department of Mathematics, University of Murcia, Campus Espinardo 15, 30100, Murcia, Spain. e-mail: [email protected] J. Orihuela, Department of Mathematics, University of Murcia, Campus Espinardo 15, 30100,· Murcia , Spain. e-mail: [email protected] M. Ruiz Galan,´ Department of Applied Math. , E. T. S. Ingenier´ıa. Edificacion.,´ c/ Severo Ochoa· s/n, 1871, Granada, Spain. -
Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations
Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations Jean Gallier and Jocelyn Quaintance Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] April 20, 2017 2 3 Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations Jean Gallier Abstract: Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, com- puter vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. One of my (selfish!) motivations in writing these notes was to understand the concept of shelling and how it is used to prove the famous Euler-Poincar´eformula (Poincar´e,1899) and the more recent Upper Bound Theorem (McMullen, 1970) for polytopes. Another of my motivations was to give a \correct" account of Delaunay triangulations and Voronoi diagrams in terms of (direct and inverse) stereographic projections onto a sphere and prove rigorously that the projective map that sends the (projective) sphere to the (projective) paraboloid works correctly, that is, maps the Delaunay triangulation and Voronoi diagram w.r.t. the lifting onto the sphere to the Delaunay diagram and Voronoi diagrams w.r.t. the traditional lifting onto the paraboloid. Here, the problem is that this map is only well defined (total) in projective space and we are forced to define the notion of convex polyhedron in projective space. -
Variational Analysis Perspective on Linear Convergence of Some First
Noname manuscript No. (will be inserted by the editor) Variational analysis perspective on linear convergence of some first order methods for nonsmooth convex optimization problems Jane J. Ye · Xiaoming Yuan · Shangzhi Zeng · Jin Zhang the date of receipt and acceptance should be inserted later Abstract We understand linear convergence of some first-order methods such as the proximal gradient method (PGM), the proximal alternating linearized minimization (PALM) algorithm and the randomized block coordinate proxi- mal gradient method (R-BCPGM) for minimizing the sum of a smooth con- vex function and a nonsmooth convex function from a variational analysis perspective. We introduce a new analytic framework based on some theories on variational analysis such as the error bound/calmness/metric subregular- ity/bounded metric subregularity. This variational analysis perspective enables us to provide some concrete sufficient conditions for checking linear conver- gence and applicable approaches for calculating linear convergence rates of these first-order methods for a class of structured convex problems where the smooth part of the objective function is a composite of a smooth strongly convex function and a linear function. In particular, we show that these con- ditions are satisfied automatically, and the modulus for the calmness/metric subregularity is practically computable in some important applications such as in the LASSO, the fused LASSO and the group LASSO. Consequently, The research was partially supported by NSERC, a general research fund from the Research Grants Council of Hong Kong, NSFC (No. 11601458) Jane J. Ye Department of Mathematics and Statistics, University of Victoria, Canada. E-mail: [email protected] Xiaoming Yuan Department of Mathematics, The University of Hong Kong, Hong Kong, China. -
A Tutorial on Convex Optimization
A Tutorial on Convex Optimization Haitham Hindi Palo Alto Research Center (PARC), Palo Alto, California email: [email protected] Abstract— In recent years, convex optimization has be- quickly deduce the convexity (and hence tractabil- come a computational tool of central importance in engi- ity) of many problems, often by inspection; neering, thanks to it’s ability to solve very large, practical 2) to review and introduce some canonical opti- engineering problems reliably and efficiently. The goal of this tutorial is to give an overview of the basic concepts of mization problems, which can be used to model convex sets, functions and convex optimization problems, so problems and for which reliable optimization code that the reader can more readily recognize and formulate can be readily obtained; engineering problems using modern convex optimization. 3) to emphasize modeling and formulation; we do This tutorial coincides with the publication of the new book not discuss topics like duality or writing custom on convex optimization, by Boyd and Vandenberghe [7], who have made available a large amount of free course codes. material and links to freely available code. These can be We assume that the reader has a working knowledge of downloaded and used immediately by the audience both linear algebra and vector calculus, and some (minimal) for self-study and to solve real problems. exposure to optimization. Our presentation is quite informal. Rather than pro- I. INTRODUCTION vide details for all the facts and claims presented, our Convex optimization can be described as a fusion goal is instead to give the reader a flavor for what is of three disciplines: optimization [22], [20], [1], [3], possible with convex optimization. -
Proximal Analysis and Boundaries of Closed Sets in Banach Space Part Ii: Applications
Can. J. Math., Vol. XXXIX, No. 2, 1987, pp. 428-472 PROXIMAL ANALYSIS AND BOUNDARIES OF CLOSED SETS IN BANACH SPACE PART II: APPLICATIONS J. M. BORWEIN AND H. M. STROJWAS Introduction. This paper is a direct continuation of the article "Proximal analysis and boundaries of closed sets in Banach space, Part I: Theory", by the same authors. It is devoted to a detailed analysis of applications of the theory presented in the first part and of its limitations. 5. Applications in geometry of normed spaces. Theorem 2.1 has important consequences for geometry of Banach spaces. We start the presentation with a discussion of density and existence of improper points (Definition 1.3) for closed sets in Banach spaces. Our considerations will be based on the "lim inf ' inclusions proven in the first part of our paper. Ti IEOREM 5.1. If C is a closed subset of a Banach space E, then the K-proper points of C are dense in the boundary of C. Proof If x is in the boundary of C, for each r > 0 we may find y £ C with \\y — 3c|| < r. Theorem 2.1 now shows that Kc(xr) ^ E for some xr e C with II* - *,ll ^ 2r. COROLLARY 5.1. ( [2] ) If C is a closed convex subset of a Banach space E, then the support points of C are dense in the boundary of C. Proof. Since for any convex set C and x G C we have Tc(x) = Kc(x) = Pc(x) = P(C - x) = WTc(x) = WKc(x) = WPc(x\ the /^-proper points of C at x, where Rc(x) is any one of the above cones, are exactly the support points. -
HP 33S Solving Trigonometry Problems the Trigonometric Functions Trigonometric Modes Practice Working Problems Involving Trig Functions
hp calculators HP 33S Solving Trigonometry Problems The trigonometric functions Trigonometric modes Practice working problems involving trig functions hp calculators HP 33S Solving Trigonometry Problems The trigonometric functions The trigonometric functions, sine, cosine, tangent, and related functions, are used in geometry, surveying, and design. They also occur in solutions to orbital mechanics, integration, and other advanced applications. The HP 33S provides the three basic functions, and their inverse, or “arc” functions. These work in degrees, radians and gradians modes. In addition, π is provided as a function on the right-shifted “cos” key, and the sign function is on the right-shifted “sin” key. The secant, cosecant and cotangent functions are easily calculated using the n, k, and q keys respectively, followed by ,. To help remember whether the secant function corresponds to the inverse sine or cosine, it can be helpful to note that the first letters of “secant” and “cosecant” are inverted in relation to those of “sine” and “cosine”, just as the secant and cosecant are the inverted cosine and sine functions. The two keys ° and ± also use trigonometric functions to convert between rectangular and radial coordinates. They can therefore be useful in some trigonometric calculations. Trigonometric modes The HP 33S can calculate trigonometric functions in any of these three modes: Degrees, Radians or Gradians. Practice working problems involving trig functions Example 1: Select the appropriate angle mode. Solution: Press the Ý key below the screen. Figure 1 Press 1, 2 or 3 to select DEGrees, RADians or GRADians mode, or use the arrow keys Ö, Õ, × and Ø to select the required mode and then press Ï.