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Lecture 5: Maxima and Minima of Convex Functions
Advanced Mathematical Programming IE417 Lecture 5 Dr. Ted Ralphs IE417 Lecture 5 1 Reading for This Lecture ² Chapter 3, Sections 4-5 ² Chapter 4, Section 1 1 IE417 Lecture 5 2 Maxima and Minima of Convex Functions 2 IE417 Lecture 5 3 Minimizing a Convex Function Theorem 1. Let S be a nonempty convex set on Rn and let f : S ! R be ¤ convex on S. Suppose that x is a local optimal solution to minx2S f(x). ² Then x¤ is a global optimal solution. ² If either x¤ is a strict local optimum or f is strictly convex, then x¤ is the unique global optimal solution. 3 IE417 Lecture 5 4 Necessary and Sufficient Conditions Theorem 2. Let S be a nonempty convex set on Rn and let f : S ! R be convex on S. The point x¤ 2 S is an optimal solution to the problem T ¤ minx2S f(x) if and only if f has a subgradient » such that » (x ¡ x ) ¸ 0 8x 2 S. ² This implies that if S is open, then x¤ is an optimal solution if and only if there is a zero subgradient of f at x¤. 4 IE417 Lecture 5 5 Alternative Optima Theorem 3. Let S be a nonempty convex set on Rn and let f : S ! R be convex on S. If the point x¤ 2 S is an optimal solution to the problem minx2S f(x), then the set of alternative optima are characterized by the set S¤ = fx 2 S : rf(x¤)T (x ¡ x¤) · 0 and rf(x¤) = rf(x) Corollaries: 5 IE417 Lecture 5 6 Maximizing a Convex Function Theorem 4. -
CORE View Metadata, Citation and Similar Papers at Core.Ac.Uk
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Bulgarian Digital Mathematics Library at IMI-BAS Serdica Math. J. 27 (2001), 203-218 FIRST ORDER CHARACTERIZATIONS OF PSEUDOCONVEX FUNCTIONS Vsevolod Ivanov Ivanov Communicated by A. L. Dontchev Abstract. First order characterizations of pseudoconvex functions are investigated in terms of generalized directional derivatives. A connection with the invexity is analysed. Well-known first order characterizations of the solution sets of pseudolinear programs are generalized to the case of pseudoconvex programs. The concepts of pseudoconvexity and invexity do not depend on a single definition of the generalized directional derivative. 1. Introduction. Three characterizations of pseudoconvex functions are considered in this paper. The first is new. It is well-known that each pseudo- convex function is invex. Then the following question arises: what is the type of 2000 Mathematics Subject Classification: 26B25, 90C26, 26E15. Key words: Generalized convexity, nonsmooth function, generalized directional derivative, pseudoconvex function, quasiconvex function, invex function, nonsmooth optimization, solution sets, pseudomonotone generalized directional derivative. 204 Vsevolod Ivanov Ivanov the function η from the definition of invexity, when the invex function is pseudo- convex. This question is considered in Section 3, and a first order necessary and sufficient condition for pseudoconvexity of a function is given there. It is shown that the class of strongly pseudoconvex functions, considered by Weir [25], coin- cides with pseudoconvex ones. The main result of Section 3 is applied to characterize the solution set of a nonlinear programming problem in Section 4. The base results of Jeyakumar and Yang in the paper [13] are generalized there to the case, when the function is pseudoconvex. -
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 -
Convex) Level Sets Integration
Journal of Optimization Theory and Applications manuscript No. (will be inserted by the editor) (Convex) Level Sets Integration Jean-Pierre Crouzeix · Andrew Eberhard · Daniel Ralph Dedicated to Vladimir Demjanov Received: date / Accepted: date Abstract The paper addresses the problem of recovering a pseudoconvex function from the normal cones to its level sets that we call the convex level sets integration problem. An important application is the revealed preference problem. Our main result can be described as integrating a maximally cycli- cally pseudoconvex multivalued map that sends vectors or \bundles" of a Eu- clidean space to convex sets in that space. That is, we are seeking a pseudo convex (real) function such that the normal cone at each boundary point of each of its lower level sets contains the set value of the multivalued map at the same point. This raises the question of uniqueness of that function up to rescaling. Even after normalising the function long an orienting direction, we give a counterexample to its uniqueness. We are, however, able to show uniqueness under a condition motivated by the classical theory of ordinary differential equations. Keywords Convexity and Pseudoconvexity · Monotonicity and Pseudomono- tonicity · Maximality · Revealed Preferences. Mathematics Subject Classification (2000) 26B25 · 91B42 · 91B16 Jean-Pierre Crouzeix, Corresponding Author, LIMOS, Campus Scientifique des C´ezeaux,Universit´eBlaise Pascal, 63170 Aubi`ere,France, E-mail: [email protected] Andrew Eberhard School of Mathematical & Geospatial Sciences, RMIT University, Melbourne, VIC., Aus- tralia, E-mail: [email protected] Daniel Ralph University of Cambridge, Judge Business School, UK, E-mail: [email protected] 2 Jean-Pierre Crouzeix et al. -
ABOUT STATIONARITY and REGULARITY in VARIATIONAL ANALYSIS 1. Introduction the Paper Investigates Extremality, Stationarity and R
1 ABOUT STATIONARITY AND REGULARITY IN VARIATIONAL ANALYSIS 2 ALEXANDER Y. KRUGER To Boris Mordukhovich on his 60th birthday Abstract. Stationarity and regularity concepts for the three typical for variational analysis classes of objects { real-valued functions, collections of sets, and multifunctions { are investi- gated. An attempt is maid to present a classification scheme for such concepts and to show that properties introduced for objects from different classes can be treated in a similar way. Furthermore, in many cases the corresponding properties appear to be in a sense equivalent. The properties are defined in terms of certain constants which in the case of regularity proper- ties provide also some quantitative characterizations of these properties. The relations between different constants and properties are discussed. An important feature of the new variational techniques is that they can handle nonsmooth 3 functions, sets and multifunctions equally well Borwein and Zhu [8] 4 1. Introduction 5 The paper investigates extremality, stationarity and regularity properties of real-valued func- 6 tions, collections of sets, and multifunctions attempting at developing a unifying scheme for defining 7 and using such properties. 8 Under different names this type of properties have been explored for centuries. A classical 9 example of a stationarity condition is given by the Fermat theorem on local minima and max- 10 ima of differentiable functions. In a sense, any necessary optimality (extremality) conditions de- 11 fine/characterize certain stationarity (singularity/irregularity) properties. The separation theorem 12 also characterizes a kind of extremal (stationary) behavior of convex sets. 13 Surjectivity of a linear continuous mapping in the Banach open mapping theorem (and its 14 extension to nonlinear mappings known as Lyusternik-Graves theorem) is an example of a regularity 15 condition. -
NMSA403 Optimization Theory – Exercises Contents
NMSA403 Optimization Theory { Exercises Martin Branda Charles University, Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Version: December 17, 2017 Contents 1 Introduction and motivation 2 1.1 Operations Research/Management Science and Mathematical Program- ming . .2 1.2 Marketing { Optimization of advertising campaigns . .2 1.3 Logistic { Vehicle routing problems . .2 1.4 Scheduling { Reparations of oil platforms . .3 1.5 Insurance { Pricing in nonlife insurance . .4 1.6 Power industry { Bidding, power plant operations . .4 1.7 Environment { Inverse modelling in atmosphere . .5 2 Introduction to optimization 6 3 Convex sets and functions 6 4 Separation of sets 8 5 Subdifferentiability and subgradient 9 6 Generalizations of convex functions 10 6.1 Quasiconvex functions . 10 6.2 Pseudoconvex functions . 12 7 Optimality conditions 13 7.1 Optimality conditions based on directions . 13 7.2 Karush{Kuhn{Tucker optimality conditions . 15 7.2.1 A few pieces of the theory . 15 7.2.2 Karush{Kuhn{Tucker optimality conditions . 17 7.2.3 Second Order Sufficient Condition (SOSC) . 19 1 Introduction and motivation 1.1 Operations Research/Management Science and Mathe- matical Programming Goal: improve/stabilize/set of a system. You can reach the goal in the following steps: • Problem understanding • Problem description { probabilistic, statistical and econometric models • Optimization { mathematical programming (formulation and solution) • Verification { backtesting, stresstesting • Implementation (Decision Support -
Monotonic Transformations: Cardinal Versus Ordinal Utility
Natalia Lazzati Mathematics for Economics (Part I) Note 10: Quasiconcave and Pseudoconcave Functions Note 10 is based on Madden (1986, Ch. 13, 14) and Simon and Blume (1994, Ch. 21). Monotonic transformations: Cardinal Versus Ordinal Utility A utility function could be said to measure the level of satisfaction associated to each commodity bundle. Nowadays, no economist really believes that a real number can be assigned to each commodity bundle which expresses (in utils?) the consumer’slevel of satisfaction with this bundle. Economists believe that consumers have well-behaved preferences over bundles and that, given any two bundles, a consumer can indicate a preference of one over the other or the indi¤erence between the two. Although economists work with utility functions, they are concerned with the level sets of such functions, not with the number that the utility function assigns to any given level set. In consumer theory these level sets are called indi¤erence curves. A property of utility functions is called ordinal if it depends only on the shape and location of a consumer’sindi¤erence curves. It is alternatively called cardinal if it also depends on the actual amount of utility the utility function assigns to each indi¤erence set. In the modern approach, we say that two utility functions are equivalent if they have the same indi¤erence sets, although they may assign di¤erent numbers to each level set. For instance, let 2 2 u : R+ R where u (x) = x1x2 be a utility function, and let v : R+ R be the utility function ! ! v (x) = u (x) + 1: These two utility functions represent the same preferences and are therefore equivalent. -
The Ekeland Variational Principle, the Bishop-Phelps Theorem, and The
The Ekeland Variational Principle, the Bishop-Phelps Theorem, and the Brøndsted-Rockafellar Theorem Our aim is to prove the Ekeland Variational Principle which is an abstract result that found numerous applications in various fields of Mathematics. As its application to Convex Analysis, we provide a proof of the famous Bishop- Phelps Theorem and some related results. Let us recall that the epigraph and the hypograph of a function f : X ! [−∞; +1] (where X is a set) are the following subsets of X × R: epi f = f(x; t) 2 X × R : t ≥ f(x)g ; hyp f = f(x; t) 2 X × R : t ≤ f(x)g : Notice that, if f > −∞ on X then epi f 6= ; if and only if f is proper, that is, f is finite at at least one point. 0.1. The Ekeland Variational Principle. In what follows, (X; d) is a metric space. Given λ > 0 and (x; t) 2 X × R, we define the set Kλ(x; t) = f(y; s): s ≤ t − λd(x; y)g = f(y; s): λd(x; y) ≤ t − sg ⊂ X × R: Notice that Kλ(x; t) = hyp[t − λ(x; ·)]. Let us state some useful properties of these sets. Lemma 0.1. (a) Kλ(x; t) is closed and contains (x; t). (b) If (¯x; t¯) 2 Kλ(x; t) then Kλ(¯x; t¯) ⊂ Kλ(x; t). (c) If (xn; tn) ! (x; t) and (xn+1; tn+1) 2 Kλ(xn; tn) (n 2 N), then \ Kλ(x; t) = Kλ(xn; tn) : n2N Proof. (a) is quite easy. -
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. -
First Order Characterizations of Pseudoconvex Functions
Serdica Math. J. 27 (2001), 203-218 FIRST ORDER CHARACTERIZATIONS OF PSEUDOCONVEX FUNCTIONS Vsevolod Ivanov Ivanov Communicated by A. L. Dontchev Abstract. First order characterizations of pseudoconvex functions are investigated in terms of generalized directional derivatives. A connection with the invexity is analysed. Well-known first order characterizations of the solution sets of pseudolinear programs are generalized to the case of pseudoconvex programs. The concepts of pseudoconvexity and invexity do not depend on a single definition of the generalized directional derivative. 1. Introduction. Three characterizations of pseudoconvex functions are considered in this paper. The first is new. It is well-known that each pseudo- convex function is invex. Then the following question arises: what is the type of 2000 Mathematics Subject Classification: 26B25, 90C26, 26E15. Key words: Generalized convexity, nonsmooth function, generalized directional derivative, pseudoconvex function, quasiconvex function, invex function, nonsmooth optimization, solution sets, pseudomonotone generalized directional derivative. 204 Vsevolod Ivanov Ivanov the function η from the definition of invexity, when the invex function is pseudo- convex. This question is considered in Section 3, and a first order necessary and sufficient condition for pseudoconvexity of a function is given there. It is shown that the class of strongly pseudoconvex functions, considered by Weir [25], coin- cides with pseudoconvex ones. The main result of Section 3 is applied to characterize the solution set of a nonlinear programming problem in Section 4. The base results of Jeyakumar and Yang in the paper [13] are generalized there to the case, when the function is pseudoconvex. The second and third characterizations are considered in Sections 5, 6. -
On Some Generalization of Convex Sets, Convex Functions, and Convex Optimization Problems
Republic of Iraq Ministry of Higher Education and Scientific Research University of Baghdad College of Education for Pure Sciences / Ibn Al-Haitham Department of Mathematics On Some Generalization of Convex Sets, Convex Functions, and Convex Optimization Problems A Thesis Submitted to the College of Education for Pure Sciences \ Ibn Al-Haitham, University of Baghdad as a Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics By Mahmood Idrees Abdulmaged Supervised By Lecturer Dr. Saba Naser Majeed 2018 AC 1439 AH ِ ِ ِ يَْرفَِع اللَّهُ الَّذي َن آمَنُوا من ُك ْم َوالَّذي َن أُوتُوا ِ ٍ الْعْلمَ دََرجَات َواللَّهُ بِمَا تَْعمَلُو َن خَبِي ر المجادلة 11 Supervisor’s Certification I certify that this thesis was prepared under my supervision at the department of mathematics, college of education for pure science / Ibn Al-Haitham, Baghdad university as a partial fulfillment of the requirements for the degree of master of science in mathematics. Signature: Name: Dr. Saba Naser Majeed Date: / /2018 In view of the available recommendation, I forward this thesis for debate by the examining committee. Signature: Name: Asst. Prof. Dr. Majeed Ahmed Weli Head of Department of Mathematics Date: / /2018 Committee Certification We certify that we have read this thesis entitled " On Some Generalization of Convex Sets, Convex Functions, and Convex Optimization Problems" and as examining committee, we examined the student (Mahmood Idrees Abdulmaged) in its contains and what is connected with it, and that in our opinion it is adequate for the partial fulfillment of the requirement for the degree of Master of Science in Mathematics. -
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.