scholarly journals Degree of pointedness of a convex function

1996 ◽  
Vol 53 (1) ◽  
pp. 159-167 ◽  
Author(s):  
Alberto Seeger
Keyword(s):  
Convex Function ◽  
Conjugate Function ◽  
Recession Cone ◽  
Effective Domain

A convex function f is said to be pointed if its epigraph has a recession cone which is pointed. Partial pointedness of f refers to the case in which such a recession cone is only partially pointed. In this note we show that the degree of pointedness of f is related to the “thickness” of the effective domain of the conjugate function f*.

Conjugates and Legendre Transforms of Convex Functions

1967 ◽  
Vol 19 ◽  
pp. 200-205 ◽  
Author(s):  
R. T. Rockafellar
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Strict Convexity ◽  
Legendre Transformation ◽  
Legendre Transform ◽  
Implicit Functions ◽  
The One ◽  
Legendre Transforms ◽  
Effective Domain ◽  
The Relationship

Fenchel's conjugate correspondence for convex functions may be viewed as a generalization of the classical Legendre correspondence, as indicated briefly in (6). Here the relationship between the two correspondences will be described in detail. Essentially, the conjugate reduces to the Legendre transform if and only if the subdifferential of the convex function is a one-to-one mapping. The one-to-oneness is equivalent to differentiability and strict convexity, plus a condition that the function become infinitely steep near boundary points of its effective domain. These conditions are shown to be the very ones under which the Legendre correspondence is well-defined and symmetric among convex functions. Facts about Legendre transforms may thus be deduced using the elegant, geometrically motivated methods of Fenchel. This has definite advantages over the more restrictive classical treatment of the Legendre transformation in terms of implicit functions, determinants, and the like.

scholarly journals On an optimal control problem for a parabolic inclusion

1996 ◽  
Vol 143 ◽  
pp. 195-217
Author(s):  
Bui an Ton
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Convex Function ◽  
Control Problem ◽  
Hilbert Spaces ◽  
Convex Subset ◽  
Lower Semi ◽  
Continuous Convex Function ◽  
Effective Domain

Let H, U be two real Hilbert spaces and let g be a proper lower semi-continuous convex function from L2 (0, T;H) into R+. For each t in [0, T], let φ(t,.) be a proper l.s.c. convex function from H into R with effective domain Dφ(t,.)) and let h be a l.s.c. convex function from a closed convex subset u of U into L2(0, T;H) withfor all u in U. The constants γ and C are positive.

Extreme-point solutions in Markov decision processes

1983 ◽  
Vol 20 (04) ◽  
pp. 835-842
Author(s):  
David Assaf
Keyword(s):  
Convex Function ◽  
Extreme Point ◽  
Markov Decision Processes ◽  
Convex Functions ◽  
Sufficient Conditions ◽  
Decision Processes ◽  
Markov Decision ◽  
Full Solution

The paper presents sufficient conditions for certain functions to be convex. Functions of this type often appear in Markov decision processes, where their maximum is the solution of the problem. Since a convex function takes its maximum at an extreme point, the conditions may greatly simplify a problem. In some cases a full solution may be obtained after the reduction is made. Some illustrative examples are discussed.

scholarly journals On Certain Differential Subordination of Harmonic Mean Related to a Linear Function

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 966
Author(s):  
Anna Dobosz ◽  
Piotr Jastrzębski ◽  
Adam Lecko
Keyword(s):  
Convex Function ◽  
Linear Function ◽  
Differential Subordination ◽  
Harmonic Mean ◽  
Best Dominant ◽  
Symmetry Properties ◽  
Differential Subordinations ◽  
Difficult Issue

In this paper we study a certain differential subordination related to the harmonic mean and its symmetry properties, in the case where a dominant is a linear function. In addition to the known general results for the differential subordinations of the harmonic mean in which the dominant was any convex function, one can study such differential subordinations for the selected convex function. In this case, a reasonable and difficult issue is to look for the best dominant or one that is close to it. This paper is devoted to this issue, in which the dominant is a linear function, and the differential subordination of the harmonic mean is a generalization of the Briot–Bouquet differential subordination.

scholarly journals A note on generalized convex functions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Syed Zaheer Ullah ◽  
Muhammad Adil Khan ◽  
Yu-Ming Chu
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Type Inequality ◽  
Generalized Convex Functions

Abstract In the article, we provide an example for a η-convex function defined on rectangle is not convex, prove that every η-convex function defined on rectangle is coordinate η-convex and its converse is not true in general, define the coordinate $(\eta _{1}, \eta _{2})$(η1,η2)-convex function and establish its Hermite–Hadamard type inequality.

scholarly journals An alternative perspective on copositive and convex relaxations of nonconvex quadratic programs

2021 ◽  
Author(s):  
E. Alper Yıldırım
Keyword(s):  
Objective Function ◽  
Convex Relaxation ◽  
Large Family ◽  
Quadratic Program ◽  
Feasible Region ◽  
Recession Cone ◽  
Convex Relaxations ◽  
Quadratic Programs ◽  
Alternative Perspective ◽  
Optimal Value

AbstractWe study convex relaxations of nonconvex quadratic programs. We identify a family of so-called feasibility preserving convex relaxations, which includes the well-known copositive and doubly nonnegative relaxations, with the property that the convex relaxation is feasible if and only if the nonconvex quadratic program is feasible. We observe that each convex relaxation in this family implicitly induces a convex underestimator of the objective function on the feasible region of the quadratic program. This alternative perspective on convex relaxations enables us to establish several useful properties of the corresponding convex underestimators. In particular, if the recession cone of the feasible region of the quadratic program does not contain any directions of negative curvature, we show that the convex underestimator arising from the copositive relaxation is precisely the convex envelope of the objective function of the quadratic program, strengthening Burer’s well-known result on the exactness of the copositive relaxation in the case of nonconvex quadratic programs. We also present an algorithmic recipe for constructing instances of quadratic programs with a finite optimal value but an unbounded relaxation for a rather large family of convex relaxations including the doubly nonnegative relaxation.

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