Disciplined Quasiconvex Programming
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Robustly Quasiconvex Function
Robustly quasiconvex function Bui Thi Hoa Centre for Informatics and Applied Optimization, Federation University Australia May 21, 2018 Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 1 / 15 Convex functions 1 All lower level sets are convex. 2 Each local minimum is a global minimum. 3 Each stationary point is a global minimizer. Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 2 / 15 Definition f is called explicitly quasiconvex if it is quasiconvex and for all λ 2 (0; 1) f (λx + (1 − λ)y) < maxff (x); f (y)g; with f (x) 6= f (y): Example 1 f1 : R ! R; f1(x) = 0; x 6= 0; f1(0) = 1. 2 f2 : R ! R; f2(x) = 1; x 6= 0; f2(0) = 0. 3 Convex functions are quasiconvex, and explicitly quasiconvex. 4 f (x) = x3 are quasiconvex, and explicitly quasiconvex, but not convex. Generalised Convexity Definition A function f : X ! R, with a convex domf , is called quasiconvex if for all x; y 2 domf , and λ 2 [0; 1] we have f (λx + (1 − λ)y) ≤ maxff (x); f (y)g: Bui Thi Hoa (CIAO, FedUni) Robustly quasiconvex function May 21, 2018 3 / 15 Example 1 f1 : R ! R; f1(x) = 0; x 6= 0; f1(0) = 1. 2 f2 : R ! R; f2(x) = 1; x 6= 0; f2(0) = 0. 3 Convex functions are quasiconvex, and explicitly quasiconvex. 4 f (x) = x3 are quasiconvex, and explicitly quasiconvex, but not convex. Generalised Convexity Definition A function f : X ! R, with a convex domf , is called quasiconvex if for all x; y 2 domf , and λ 2 [0; 1] we have f (λx + (1 − λ)y) ≤ maxff (x); f (y)g: Definition f is called explicitly quasiconvex if it is quasiconvex and -
Local Maximum Points of Explicitly Quasiconvex Functions
Local maximum points of explicitly quasiconvex functions Item Type Article Authors Bagdasar, Ovidiu; Popovici, Nicolae Citation Bagdasar, O. & Popovici, (2015) 'Local maximum points of explicitly quasiconvex functions' Optimization Letters, 9: 769. doi:10.1007/s11590-014-0781-3 DOI 10.1007/s11590-014-0781-3 Publisher Springer Journal Optimization Letters Rights Archived with thanks to Optimization Letters Download date 25/09/2021 20:26:30 Link to Item http://hdl.handle.net/10545/620886 Optimization Letters manuscript No. (will be inserted by the editor) Local maximum points of explicitly quasiconvex functions Ovidiu Bagdasar · Nicolae Popovici Received: date / Accepted: date Abstract This work concerns (generalized) convex real-valued functions defined on a nonempty convex subset of a real topological linear space. Its aim is twofold. The first concerns explicitly quasiconvex functions. As a counterpart of some known results, it is shown that any local maximum point of such a function is actually a global minimum point whenever it belongs to the intrinsic core of the function's domain. Secondly, we establish a new characterization of strictly convex normed spaces by applying this property for a particular class of convex functions. Keywords Local maximum point · Relative algebraic interior · Convex function · Explicitly quasiconvex function · Strictly convex space · Least squares problem 1 Introduction Optimization problems involving explicitly quasiconvex objective functions, i.e., real-valued functions which are both quasiconvex and semistrictly quasiconvex, have been intensively studied in the literature, beginning with the pioneering works by Martos [6] and Karamardian [5]. These functions are of special interest since they preserve several fundamental properties of convex functions. -
A Class of Functions That Are Quasiconvex but Not Polyconvex
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 12-2003 A Class of Functions That Are Quasiconvex But Not Polyconvex Catherine S. Remus University of Tennessee - Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Mathematics Commons Recommended Citation Remus, Catherine S., "A Class of Functions That Are Quasiconvex But Not Polyconvex. " Master's Thesis, University of Tennessee, 2003. https://trace.tennessee.edu/utk_gradthes/2219 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Catherine S. Remus entitled "A Class of Functions That Are Quasiconvex But Not Polyconvex." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Mathematics. Henry C. Simpson, Major Professor We have read this thesis and recommend its acceptance: Charles Collins, G. Samuel Jordan Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewith a thesis written by Catherine S. Remus entitled “A Class of Functions That Are Quasiconvex But Not Polyconvex”. -
Characterisations of Quasiconvex Functions
BULL. AUSTRAL. MATH. SOC. 26B25, 49J52 VOL. 48 (1993) [393-406] CHARACTERISATIONS OF QUASICONVEX FUNCTIONS DINH THE LUC In this paper we introduce the concept of quasimonotone maps and prove that a lower semicontinuous function on an infinite dimensional space is quasiconvex if and only if its generalised subdifferential or its directional derivative is quasimonotone. 1. INTRODUCTION As far as we know quasiconvex functions were first mentioned by de Finetti in his work "Sulle Straficazioni Convesse" in 1949, [6]. Since then much effort has been focused on the study of this class of functions for they have much in common with convex functions and they are used in several areas of science including mathematics, operations research, economics et cetera (see [15]). One of the most important properties of convex functions is that their level sets are convex. This property is also a fundamental geometric characterisation of quasiconvex functions which sometimes is treated as their definition. However, the most attractive characterisations of quasiconvex functions are those which involve gradients (a detailed account of the current state of research on the topic can be found in [l]). As to generalised derivatives of quasiconvex functions, very few results exist (see [1,7]). In [7], a study of quasiconvex functions is presented via Clarke's subdifferential, but the authors restrict themselves to the case of Lipschitz functions on a finite dimensional space only. The aim of our paper is to characterise lower semicontinuous quasiconvex functions in terms of generalised (Clarke-Rockafellar) subdifferentials and directional derivatives in infinite dimensional spaces. Namely, we shall first introduce a new concept of quasi- monotone maps; and then we prove the equivalence between the quasiconvexity of a function and the quasimonotonicity of its generalised subdifferential and its directional derivative. -
Subdifferential Properties of Quasiconvex and Pseudoconvex Functions: Unified Approach1
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 97, No. I, pp. 29-45, APRIL 1998 Subdifferential Properties of Quasiconvex and Pseudoconvex Functions: Unified Approach1 D. AUSSEL2 Communicated by M. Avriel Abstract. In this paper, we are mainly concerned with the characteriza- tion of quasiconvex or pseudoconvex nondifferentiable functions and the relationship between those two concepts. In particular, we charac- terize the quasiconvexity and pseudoconvexity of a function by mixed properties combining properties of the function and properties of its subdifferential. We also prove that a lower semicontinuous and radially continuous function is pseudoconvex if it is quasiconvex and satisfies the following optimality condition: 0sdf(x) =f has a global minimum at x. The results are proved using the abstract subdifferential introduced in Ref. 1, a concept which allows one to recover almost all the subdiffer- entials used in nonsmooth analysis. Key Words. Nonsmooth analysis, abstract subdifferential, quasicon- vexity, pseudoconvexity, mixed property. 1. Preliminaries Throughout this paper, we use the concept of abstract subdifferential introduced in Aussel et al. (Ref. 1). This definition allows one to recover almost all classical notions of nonsmooth analysis. The aim of such an approach is to show that a lot of results concerning the subdifferential prop- erties of quasiconvex and pseudoconvex functions can be proved in a unified way, and hence for a large class of subdifferentials. We are interested here in two aspects of convex analysis: the charac- terization of quasiconvex and pseudoconvex lower semicontinuous functions and the relationship between these two concepts. 1The author is indebted to Prof. M. Lassonde for many helpful discussions leading to a signifi- cant improvement in the presentation of the results. -
Chapter 2 Convex Functions
Chapter 2 Convex functions 2.1 Basic properties and examples 2.1.1 Definition A function f : Rn → R is convex if dom f is a convex set and if for all x, y ∈ dom f,and θ with 0 ≤ θ ≤ 1, we have f(θx +(1− θ)y) ≤ θf(x)+(1− θ)f(y). (2.1) Geometrically, this inequality means that the line segment between (x, f(x)) and (y, f(y)) (i.e.,thechord from x to y) lies above the graph of f (figure 2.1). A function f is strictly convex if strict inequality holds in (2.1) whenever x = y and 0 <θ<1. We say f is concave if −f is convex, and strictly concave if −f is strictly convex. For an affine function we always have equality in (2.1), so all affine (and therefore also linear) functions are both convex and concave. Conversely, any function that is convex and concaveisaffine. A function is convex if and only if it is convex when restricted to any line that intersects its domain. In other words f is convex if and only if for all x ∈ dom f and all v, the function (y, f(y)) (x, f(x)) Figure 2.1: Graph of a convex function. The chord (i.e., line segment) between any two points on the graph lies above the graph. 43 44 CHAPTER 2. CONVEX FUNCTIONS h(t)=f(x + tv) is convex (on its domain, {t | x + tv ∈ dom f}). This property is very useful, since it allows us to check whether a function is convex by restrictingit to a line. -
Efficient Algorithms for Solving Nonlinear Fractional Programming
Filomat 33:7 (2019), 2149–2179 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL1907149D University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat Efficient Algorithms for Solving Nonlinear Fractional Programming Problems Azam Dolatnezhadsomarina, Esmaile Khorrama, Latif Pourkarimib aDepartment of Mathematics and Computer Sciences, Amirkabir University of Technology, Tehran, Iran bDepartment of Mathematics, Faculty of Sciences, Razi University, Kermanshah, Iran Abstract. In this paper, an efficient algorithm based on the Pascoletti-Serafini scalarization (PS) approach is proposed to obtain almost uniform approximations of the entire Pareto front of bi-objective optimization problems. Five test problems with convex, non-convex, connected, and disconnected Pareto fronts are ap- plied to evaluate the quality of approximations obtained by the proposed algorithm. Results are compared with results of some algorithms including the normal constraint (NC), weighted constraint (WC), Ben- son type, differential evolution (DE) with binomial crossover, non-dominated sorting genetic algorithm-II (NSGA-II), and S metric selection evolutionary multiobjective algorithm (SMS-EMOA). The results confirm the effectiveness of the presented bi-objective algorithm in terms of the quality of approximations of the Pareto front and CPU time. In addition, two algorithms are presented for approximately solving fractional programming (FP) problems. The first algorithm is based on an objective space cut and bound method for solving convex FP problems and the second algorithm is based on the proposed bi-objective algorithm for solving nonlinear FP problems. In addition, several examples are provided to demonstrate the performance of these suggested fractional algorithms. 1. Introduction A nonlinear fractional programming (FP) problem minimizes or maximizes a ratio of two functions subject to given constraints. -
A New Algorithm for Generalized Fractional Programs 1
Mathematical Programming 72 (1996) 147-175 A new algorithm for generalized fractional programs 1 A.I. Barros a, J.B.G. Frenk ~'* , S. Schaible c, S. Zhang b a D.E.I.O.. FacuMade de CiOncias, Uniuersi~hule de Lisboa. Campo Grande. 1700 Lisboa. Portugal b Econometrtc.... lnstttute. Erasmus Umuerstty. P.O. Box 1738. 3000 DR Rotterdam. The Netherlands c University of Cal([ornia at Riverside. Riverside, CA 92521, USA Received 2 February 1994 Abstract A new dual problem for convex generalized fractional programs with no duality gap is presented and it is shown how this dual problem can be efficiently solved using a parametric approach. The resulting algorithm can be seen as "'dual'" to the Dinkelbach-type algorithm for generalized fractional programs since it approximates the optimal objective value of the dual (primal) problem from below. Convergence results for this algorithm are derived and an easy condition to achieve superlinear convergence is also established. Moreover, under some additional assumptions the algorithm also recovers at the same time an optimal solution of the primal problem. We also consider a variant of this new algorithm, based on scaling the "'dual" parametric function, The numerical results, in case of quadratic-linear ratios and linear constraints, show that the performance of the new algorithm and its scaled version is superior to that of the Dinkelbach-type algorithms. From the computational results it also appears that contrary to the primal approach, the "'dual" approach is less influenced by scaling. Keywords: Fractional programming; Generalized fractional programming; Dinkelbach-type algorithms: Quasi- convexity; Karush-Kuhn-Tucker conditions: Duality 1. -
Glimpses Upon Quasiconvex Analysis Jean-Paul Penot
Glimpses upon quasiconvex analysis Jean-Paul Penot To cite this version: Jean-Paul Penot. Glimpses upon quasiconvex analysis. 2007. hal-00175200 HAL Id: hal-00175200 https://hal.archives-ouvertes.fr/hal-00175200 Preprint submitted on 27 Sep 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ESAIM: PROCEEDINGS, Vol. ?, 2007, 1-10 Editors: Will be set by the publisher DOI: (will be inserted later) GLIMPSES UPON QUASICONVEX ANALYSIS Jean-Paul Penot Abstract. We review various sorts of generalized convexity and we raise some questions about them. We stress the importance of some special subclasses of quasiconvex functions. Dedicated to Marc Att´eia R´esum´e. Nous passons en revue quelques notions de convexit´eg´en´eralis´ee.Nous tentons de les relier et nous soulevons quelques questions. Nous soulignons l’importance de quelques classes particuli`eres de fonctions quasiconvexes. 1. Introduction Empires usually are well structured entities, with unified, strong rules (for instance, the length of axles of carts in the Chinese Empire and the Roman Empire, a crucial rule when building a road network). On the contrary, associated kingdoms may have diverging rules and uses. -
CONVEX SETS and CONCAVE FUNCTIONS W. Erwin Diewert
1 ECONOMICS 594: LECTURE NOTES CHAPTER 2: CONVEX SETS AND CONCAVE FUNCTIONS W. Erwin Diewert January 31, 2008. 1. Introduction Many economic problems have the following structure: (i) a linear function is minimized subject to a nonlinear constraint; (ii) a linear function is maximized subject to a nonlinear constraint or (iii) a nonlinear function is maximized subject to a linear constraint. Examples of these problems are: (i) the producer’s cost minimization problem (or the consumer’s expenditure minimization problem); (ii) the producer’s profit maximization problem and (iii) the consumer’s utility maximization problem. These three constrained optimization problems play a key role in economic theory. In each of the above 3 problems, linear functions appear in either the objective function (the function being maximized or minimized) or the constraint function. If we are N T maximizing or minimizing the linear function of x, say n=1 pnxn p x, where p 1 (p1,…,pN) is a vector of prices and x (x1,…,xN) is a vector of decision variables, then after the optimization problem is solved, the optimized objective function can be regarded as a function of the price vector, say G(p), and perhaps other variables that appear in the constraint function. Let F(x) be the nonlinear function which appears in the constraint in problems (i) and (ii). Then under certain conditions, the optimized objective function G(p) can be used to reconstruct the nonlinear constraint function F(x). This correspondence between F(x) and G(p) is known as duality theory. In the following chapter, we will see how the use of dual functions can greatly simplify economic modeling. -
Generalized Convexity and Optimization
Lecture Notes in Economics and Mathematical Systems 616 Generalized Convexity and Optimization Theory and Applications Bearbeitet von Alberto Cambini, Laura Martein 1. Auflage 2008. Taschenbuch. xiv, 248 S. Paperback ISBN 978 3 540 70875 9 Format (B x L): 15,5 x 23,5 cm Gewicht: 405 g Wirtschaft > Betriebswirtschaft: Theorie & Allgemeines > Wirtschaftsmathematik und - statistik Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. 2 Non-Differentiable Generalized Convex Functions 2.1 Introduction In several economic models convexity appears to be a restrictive condition. For instance, classical assumptions in Economics include the convexity of the production set in producer theory and the convexity of the upper level sets of the utility function in consumer theory. On the other hand, in Optimization it is important to know when a local minimum (maximum) is also global. Such a useful property is not exclusive to convexity (concavity). All this has led to the introduction of new classes of functions which have con- vex lower/upper level sets and which verify the local-global property, starting with the pioneer work of Arrow–Enthoven [7]. In this chapter we shall introduce the class of quasiconvex, strictly quasicon- vex, and semistrictly quasiconvex functions and the inclusion relationships between them are studied. For functions in one variable, a complete charac- terization of the new classes is established. -
Risk- and Ambiguity-Averse Portfolio Optimization with Quasiconcave Utility Functionals
Finance Stoch (2017) 21:397–425 DOI 10.1007/s00780-016-0318-y Risk- and ambiguity-averse portfolio optimization with quasiconcave utility functionals Sigrid Källblad1 Received: 25 November 2014 / Accepted: 15 February 2016 / Published online: 15 March 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com Abstract Motivated by recent axiomatic developments, we study the risk- and ambi- guity-averse investment problem where trading takes place in continuous time over a fixed finite horizon and terminal payoffs are evaluated according to criteria defined in terms of quasiconcave utility functionals. We extend to the present setting certain ex- istence and duality results established for so-called variational preferences by Schied (Finance Stoch. 11:107–129, 2007). The results are proved by building on existing re- sults for the classical utility maximization problem, combined with a careful analysis of the involved quasiconvex and semicontinuous functions. Keywords Model uncertainty · Ambiguity aversion · Quasiconvex risk measures · Optimal investment · Robust portfolio selection · Duality theory Mathematics Subject Classification (2010) 49N15 · 62C20 · 91B06 · 91B16 · 91G10 · 91G80 JEL Classification G11 · C61 · D81 1 Introduction The optimal investment problem of choosing the best way to allocate investors’ cap- ital is often formulated as the problem of maximizing, over admissible investment strategies, the expected utility of terminal wealth. The formulation relies on the ax- iomatic foundation developed by von Neumann and Morgenstern [77] and Savage B S. Källblad [email protected] 1 Institut für Stochastik und Wirtschaftsmathematik, Technische Universität Wien, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria 398 S. Källblad [70]. In continuous-time optimal portfolio selection, the study dates back to the sem- inal contributions of Merton [60, 61].