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Extreme point

About: Extreme point is a research topic. Over the lifetime, 2675 publications have been published within this topic receiving 49361 citations. The topic is also known as: Extreme points.


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Book
03 Mar 1993
TL;DR: The book is a solid reference for professionals as well as a useful text for students in the fields of operations research, management science, industrial engineering, applied mathematics, and also in engineering disciplines that deal with analytical optimization techniques.
Abstract: COMPREHENSIVE COVERAGE OF NONLINEAR PROGRAMMING THEORY AND ALGORITHMS, THOROUGHLY REVISED AND EXPANDED"Nonlinear Programming: Theory and Algorithms"--now in an extensively updated Third Edition--addresses the problem of optimizing an objective function in the presence of equality and inequality constraints. Many realistic problems cannot be adequately represented as a linear program owing to the nature of the nonlinearity of the objective function and/or the nonlinearity of any constraints. The "Third Edition" begins with a general introduction to nonlinear programming with illustrative examples and guidelines for model construction.Concentration on the three major parts of nonlinear programming is provided: Convex analysis with discussion of topological properties of convex sets, separation and support of convex sets, polyhedral sets, extreme points and extreme directions of polyhedral sets, and linear programmingOptimality conditions and duality with coverage of the nature, interpretation, and value of the classical Fritz John (FJ) and the Karush-Kuhn-Tucker (KKT) optimality conditions; the interrelationships between various proposed constraint qualifications; and Lagrangian duality and saddle point optimality conditionsAlgorithms and their convergence, with a presentation of algorithms for solving both unconstrained and constrained nonlinear programming problemsImportant features of the "Third Edition" include: New topics such as second interior point methods, nonconvex optimization, nondifferentiable optimization, and moreUpdated discussion and new applications in each chapterDetailed numerical examples and graphical illustrationsEssential coverage of modeling and formulating nonlinear programsSimple numerical problemsAdvanced theoretical exercisesThe book is a solid reference for professionals as well as a useful text for students in the fields of operations research, management science, industrial engineering, applied mathematics, and also in engineering disciplines that deal with analytical optimization techniques. The logical and self-contained format uniquely covers nonlinear programming techniques with a great depth of information and an abundance of valuable examples and illustrations that showcase the most current advances in nonlinear problems.

6,259 citations

Journal ArticleDOI
TL;DR: In this paper, the authors investigated pure strategy sequential equilibria of repeated games with imperfect monitoring, and they showed that the latter include solutions having a "bang-bang" property, which affords a significant simplification of the equilibrium that need be considered.
Abstract: This paper investigates pure strategy sequential equilibria of repeated games with imperfect monitoring. The approach emphasizes the equilibrium value set and the static optimization problems embedded in extremal equilibria. A succession of propositions, central among which is "self-generation," allow properties of constrained efficient supergame equilibria to be deduced from the solutions of the static problems. We show that the latter include solutions having a "bang-bang" property; this affords a significant simplification of the equilibria that need be considered. These results apply to a broad class of asymmetric games, thereby generalizing our earlier work on optimal cartel equilibria. The bang-bang theorem is strengthened to a necessity result: under certain conditions, efficient sequential equilibria have the property that after every history, the value to players of the remainder of the equilibrium must be an extreme point of the equilibrium value set. General implications of the self-generation and bang-bang propositions include a proof of the monotonicity of the equilibrium average value set in the discount factor, and an iterative procedure for computing the value set.

1,013 citations

Journal ArticleDOI
TL;DR: In this paper, simple constructive proofs are given of solutions to the matric matric system Mz − ω = q; z ≧ 0; ω ≧ 1; zT = 0, for various kinds of data M, q, which embrace quadratic programming and the problem of finding equilibrium points of bimatrix games.
Abstract: Some simple constructive proofs are given of solutions to the matric system Mz − ω = q; z ≧ 0; ω ≧ 0; and zT ω = 0, for various kinds of data M, q, which embrace the quadratic programming problem and the problem of finding equilibrium points of bimatrix games. The general scheme is, assuming non-degeneracy, to generate an adjacent extreme point path leading to a solution. The scheme does not require that some functional be reduced.

966 citations

Journal ArticleDOI
TL;DR: It is proved that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unlessP = NP, and a complexity classification for all special cases with a fixed number of processing times is obtained.
Abstract: We consider the following scheduling problem. There arem parallel machines andn independent jobs. Each job is to be assigned to one of the machines. The processing of jobj on machinei requires timep ij . The objective is to find a schedule that minimizes the makespan. Our main result is a polynomial algorithm which constructs a schedule that is guaranteed to be no longer than twice the optimum. We also present a polynomial approximation scheme for the case that the number of machines is fixed. Both approximation results are corollaries of a theorem about the relationship of a class of integer programming problems and their linear programming relaxations. In particular, we give a polynomial method to round the fractional extreme points of the linear program to integral points that nearly satisfy the constraints. In contrast to our main result, we prove that no polynomial algorithm can achieve a worst-case ratio less than 3/2 unlessP = NP. We finally obtain a complexity classification for all special cases with a fixed number of processing times.

953 citations

Book
01 Jan 1971
TL;DR: In this paper, the authors present a representation of points by boundary measures on a compact convex set, and define a set of simplex with closed extreme boundary measures as a Choquet boundary.
Abstract: I Representations of Points by Boundary Measures.- 1. Distinguished Classes of Functions on a Compact Convex Set.- Classes of continuous and semicontinuous, affine and convex functions.- Uniform and pointwise approximation theorems.-Envelopes.-*Grothendieck's completeness theorem.-Theorems of Banach-Dieudonne and Krein-Smulyan*.- 2. Weak Integrals, Moments and Barycenters.- Preliminaries and notations from integration theory.-An existence theorem for weak integrals.-Vague density of point-measures with prescribed barycenter.-*Choquet's barycenter formula for affine Baire functions of first class, and a counterexample for affine functions of higher class*.- 3. Comparison of Measures on a Compact Convex Set.- Ordering of measures.-The concept of dilation for simple measures.-The fundamental lemma on the existence of majorants.-Characterization of envelopes by integrals.-*Dilation of general measures.-Cartier's Theorem*.- 4. Choquet's Theorem.- A characterization of extreme points by means of envelopes.-The concept of a boundary set.-Herve's theorem on the existence of a strictly convex function on a metrizable compact convex set.-The concept of a boundary measure, and Mokobodzki's characterization of boundary measures.-The integral representation theorem of Choquet and Bishop - de Leeuw.-A maximum principle for superior limits of 1.s.c. convex functions.-Bishop - de Leeuw's integral theorem relatively to a ?-field on the extreme boundary.-*A counterexample based on the "porcupine topology"*.- 5. Abstract Boundaries Defined by Cones of Functions.- The concept of a Choquet boundary.-Bauer's maximum principle.-The Choquet-Edwards theorem that Choquet boundaries are Baire spaces.-The concept of a Silov boundary.-Integral representation by means of measures on the Choquet boundary.- 6. Unilateral Representation Theorems with Application to Simplicial Boundary Measures.- Ordered convex compacts.-Existence of maximal extreme points.-Characterization of the set of maximal extreme points as a Choquet boundary.-Definition and basic properties of simplicial measures.-Existence of simplicial boundary measures, and the Caratheodory Theorem in ?n.-Decomposition of representing boundary measures into simplicial components.- II Structure of Compact Convex Sets.- 1. Order-unit and Base-norm Spaces.- Basic properties of (Archimedean) order-unit spaces.-A representation theorem of Kadison.-The vector-lattice theorem of Stone-Kakutani-KreinYosida.-Duality of order-unit and base-norm spaces.- 2. Elementary Embedding Theorems.- Representation of a closed subspace A of C?(X) as an A(K)-space by the canonical embedding of X in A*.-The concept of an "abstract compact convex" and its regular embedding in a locally convex Hausdorff space.-The connection between compact convex sets and locally compact cones.- 3. Choquet Simplexes.- Riesz' decomposition property and lattice cones.-Choquet's uniqueness theorem.-Choquet-Meyer's characterizations of simplexes by envelopes.-Edward's separation theorem.-Continuous affine extensions of functions defined on compact subsets of the extreme boundary of a simplex.-Affine Borel extensions of functions defined on the extreme boundary of a simplex.-*Examples of "non-metrizable" pathologies in simplexes.*.- 4. Bauer Simplexes and the Dirichlet Problem of the Extreme Boundary.- Bauer's characterizations of simplexes with closed extreme boundary.-The Dirichlet problem of the extreme boundary.-A criterion for the existence of continuous affine extensions of maps defined on extreme boundaries.- 5. Order Ideals, Faces, and Parts.- Elementary properties of order ideals and faces.-Extension property and characteristic number.-Archimedean and strongly Archimedean ideals and faces.-Exposed and relatively exposed faces.-Specialization to simplexes.-The concept of a "part", and an inequality of Harnack type.-Characterization of the parts of a simplex in terms of representing measures.-*An example of an Archimedean face which is not strongly Archimedean.*.- 6. Split-faces and Facial Topology.- Definition and elementary properties of split faces.-Characterization of split faces by relativization of orthogonal measures.-An extension theorem for continuous affine functions defined on a split face.- The facial topology.-Specialization to simplexes.-*Near-lattice ideals, and primitive ideal space.-The connection between facial topology and hull kernel topology.-Compact convex sets with sufficiently many inner automorphisms.-A remark on the applications to C*-algebras.*.- 7. The Concept of Center for A(K).- Extension of facially continuous functions.-The facial topology is Hausdorff for Bauer simplexes only.-The concept of center, and the connections with facially continuous functions and order-bounded operators.-Convex compact sets with trivial center.-*An example of a prime simplex.-Stormer's characterization of Bauer simplexes.*.- 8. Existence and Uniqueness of Maximal Central Measures Representing Points of an Arbitrary Compact Convex Set.- The relation xoy, and the concept of a primary point.-*.- 5. Abstract Boundaries Defined by Cones of Functions.- The concept of a Choquet boundary.-Bauer's maximum principle.-The Choquet-Edwards theorem that Choquet boundaries are Baire spaces.-The concept of a Silov boundary.-Integral representation by means of measures on the Choquet boundary.- 6. Unilateral Representation Theorems with Application to Simplicial Boundary Measures.- Ordered convex compacts.-Existence of maximal extreme points.-Characterization of the set of maximal extreme points as a Choquet boundary.-Definition and basic properties of simplicial measures.-Existence of simplicial boundary measures, and the Caratheodory Theorem in ?n.-Decomposition of representing boundary measures into simplicial components.- II Structure of Compact Convex Sets.- 1. Order-unit and Base-norm Spaces.- Basic properties of (Archimedean) order-unit spaces.-A representation theorem of Kadison.-The vector-lattice theorem of Stone-Kakutani-KreinYosida.-Duality of order-unit and base-norm spaces.- 2. Elementary Embedding Theorems.- Representation of a closed subspace A of C?(X) as an A(K)-space by the canonical embedding of X in A*.-The concept of an "abstract compact convex" and its regular embedding in a locally convex Hausdorff space.-The connection between compact convex sets and locally compact cones.- 3. Choquet Simplexes.- Riesz' decomposition property and lattice cones.-Choquet's uniqueness theorem.-Choquet-Meyer's characterizations of simplexes by envelopes.-Edward's separation theorem.-Continuous affine extensions of functions defined on compact subsets of the extreme boundary of a simplex.-Affine Borel extensions of functions defined on the extreme boundary of a simplex.-*Examples of "non-metrizable" pathologies in simplexes.*.- 4. Bauer Simplexes and the Dirichlet Problem of the Extreme Boundary.- Bauer's characterizations of simplexes with closed extreme boundary.-The Dirichlet problem of the extreme boundary.-A criterion for the existence of continuous affine extensions of maps defined on extreme boundaries.- 5. Order Ideals, Faces, and Parts.- Elementary properties of order ideals and faces.-Extension property and characteristic number.-Archimedean and strongly Archimedean ideals and faces.-Exposed and relatively exposed faces.-Specialization to simplexes.-The concept of a "part", and an inequality of Harnack type.-Characterization of the parts of a simplex in terms of representing measures.-*An example of an Archimedean face which is not strongly Archimedean.*.- 6. Split-faces and Facial Topology.- Definition and elementary properties of split faces.-Characterization of split faces by relativization of orthogonal measures.-An extension theorem for continuous affine functions defined on a split face.- The facial topology.-Specialization to simplexes.-*Near-lattice ideals, and primitive ideal space.-The connection between facial topology and hull kernel topology.-Compact convex sets with sufficiently many inner automorphisms.-A remark on the applications to C*-algebras.*.- 7. The Concept of Center for A(K).- Extension of facially continuous functions.-The facial topology is Hausdorff for Bauer simplexes only.-The concept of center, and the connections with facially continuous functions and order-bounded operators.-Convex compact sets with trivial center.-*An example of a prime simplex.-Stormer's characterization of Bauer simplexes.*.- 8. Existence and Uniqueness of Maximal Central Measures Representing Points of an Arbitrary Compact Convex Set.- The relation xoy, and the concept of a primary point.-A point x is primary iff the local center at x is trivial.-The concept of a central measure.- Existence and uniqueness of maximal central measures in a special case.-The "lifting" technique.-Wils' theorem on the existence and uniqueness of maximal central measures which are pseudo-carried by the set of primary points.- References.

849 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202329
202288
2021104
2020121
2019127
201898