scholarly journals A duality theorem for nondifferentiable convex programming with operatorial constraints

1980 ◽  
Vol 22 (1) ◽  
pp. 145-152 ◽  
Author(s):  
P. Kanniappan ◽  
Sundaram M.A. Sastry
Keyword(s):  
Convex Function ◽  
Objective Function ◽  
Programming Problem ◽  
Duality Theorem ◽  
Topological Linear Space ◽  
Lower Semi ◽  
Convex Objective Function ◽  
Continuous Convex Function ◽  
Topological Linear ◽  
Locally Convex

A duality theorem of Wolfe for non-linear differentiable programming is now extended to minimization of a non-differentiable, convex, objective function defined on a general locally convex topological linear space with a non-differentiable operatorial constraint, which is regularly subdifferentiable. The gradients are replaced by subgradients. This extended duality theorem is then applied to a programming problem where the objective function is the sum of a positively homogeneous, lower semi continuous, convex function and a subdifferentiable, convex function. We obtain another duality theorem which generalizes a result of Schechter.

scholarly journals Duality theorems and an optimality condition for non-differentiable convex programming

1982 ◽  
Vol 32 (3) ◽  
pp. 369-379 ◽  
Author(s):  
P. Kanniappan ◽  
Sundaram M. A. Sastry
Keyword(s):  
Convex Function ◽  
Objective Function ◽  
Programming Problem ◽  
Convex Programming ◽  
Duality Theorem ◽  
Sufficient Optimality Conditions ◽  
Convex Programming Problem ◽  
Lower Semi ◽  
Continuous Convex Function ◽  
Necessary And Sufficient

AbstractNecessary and sufficient optimality conditions of Kuhn-Tucker type for a convex programming problem with subdifferentiable operator constraints have been obtained. A duality theorem of Wolfe's type has been derived. Assuming that the objective function is strictly convex, a converse duality theorem is obtained. The results are then applied to a programming problem in which the objective function is the sum of a positively homogeneous, lower-semi-continuous, convex function and a continuous convex function.

scholarly journals A Fenchel-Rockafellar type duality theorem for maximization

1979 ◽  
Vol 20 (2) ◽  
pp. 193-198 ◽  
Author(s):  
Ivan Singer
Keyword(s):  
Convex Space ◽  
Duality Theorem ◽  
Locally Convex Space ◽  
Bounded Subset ◽  
Convex Functional ◽  
Lower Semi ◽  
Locally Convex

We prove that sup(f-h)(E) = sup(h*-f*)(E*), where f is a proper lower semi-continuous convex functional on a real locally convex space E, h: E → = [-∞, +∞] is an arbitrary-functional and, f*, h* are their convex conjugates respectively. When h = δG, the indicator of a bounded subset G of E, this yields a formula for sup f(G).

The graph-theoretic analogue of Tietze's characterization of a convex set and its generalization

1972 ◽  
Vol 72 (1) ◽  
pp. 7-9
Author(s):  
M. D. Guay
Keyword(s):  
Linear Space ◽  
Closed Subset ◽  
Convex Sets ◽  
Convex Set ◽  
Topological Linear Space ◽  
Connected Subset ◽  
Graph Theoretic ◽  
Topological Linear ◽  
Locally Convex

Introduction. One of the most satisfying theorems in the theory of convex sets states that a closed connected subset of a topological linear space which is locally convex is convex. This was first established in En by Tietze and was later extended by other authors (see (3)) to a topological linear space. A generalization of Tietze's theorem which appears in (2) shows if S is a closed subset of a topological linear space such that the set Q of points of local non-convexity of S is of cardinality n < ∞ and S ~ Q is connected, then S is the union of n + 1 or fewer convex sets. (The case n = 0 is Tietze's theorem.)

scholarly journals A programming problem with an Lp norm in the objective function

1976 ◽  
Vol 19 (3) ◽  
pp. 333-342 ◽  
Author(s):  
Bertram Mond ◽  
Murray Schechter
Keyword(s):  
Objective Function ◽  
Programming Problem ◽  
Differentiable Function ◽  
Duality Theorem ◽  
Quadratic Function ◽  
Inequality Constraints ◽  
Square Root ◽  
Lp Norm ◽  
Differentiable Functions ◽  
Converse Duality Theorem

AbstractWe consider a programming problem in which the objective function is the sum of a differentiable function and the p norm of Sx, where S is a matrix and p > 1. The constraints are inequality constraints defined by differentiable functions. With the aid of a recent transposition theorem of Schechter we get a duality theorem and also a converse duality theorem for this problem. This result generalizes a result of Mond in which the objective function contains the square root of a positive semi-definite quadratic function.

A Tauberian theorem for statistical convergence

1989 ◽  
Vol 106 (2) ◽  
pp. 277-280 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Linear Space ◽  
Tauberian Theorem ◽  
Statistical Convergence ◽  
Slow Oscillation ◽  
Topological Linear Space ◽  
Tauberian Condition ◽  
Topological Linear ◽  
Locally Convex

The notion of statistical convergence of a sequence (xk) in a locally convex Hausdorff topological linear space X was introduced recently by Maddox[5], where it was shown that the slow oscillation of (sk) was a Tauberian condition for the statistical convergence of (sk).

scholarly journals Duality theorems for convex programming without constraint qualification

1984 ◽  
Vol 36 (2) ◽  
pp. 253-266 ◽  
Author(s):  
P. Kanniappan
Keyword(s):  
Convex Function ◽  
Convex Programming ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Convex Programming Problem ◽  
Duality Theorems ◽  
Finite Dimensional ◽  
Lower Semi ◽  
Continuous Convex Function

AbstractInvoking a recent characterization of Optimality for a convex programming problem with finite dimensional range without any constraint qualification given by Borwein and Wolkowicz, we establish duality theorems. These duality theorems subsume numerous earlier duality results with constraint qualifications. We apply our duality theorems in the case of the objective function being the sum of a positively homogeneous, lower-semi-continuous, convex function and a subdifferentiable convex function. We also study specific problems of the above type in this setting.

scholarly journals Multipliers and operators on the tempered ultradistribution spaces of Roumieu type for the distributional Hankel-type transformation spaces

2006 ◽  
Vol 2006 ◽  
pp. 1-7
Author(s):  
P. K. Banerji ◽  
S. K. Al-Omari
Keyword(s):  
Linear Space ◽  
Topological Linear Space ◽  
Type Transformation ◽  
Topological Linear ◽  
Locally Convex

The tempered ultradistribution space of Roumieu type for the spaceHμ,νis defined, which is a subspace of the Hausdörff locally convex topological linear space. Further, results are obtained for the multipliers and operators on the tempered ultradistribution spaces for the distributional Hankel-type transformation spaces.

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.

scholarly journals Tilings of Normed Spaces

2017 ◽  
Vol 69 (02) ◽  
pp. 321-337 ◽  
Author(s):  
Carlo Alberto De Bernardi ◽  
Libor Veselý
Keyword(s):  
Pairwise Disjoint ◽  
Convex Sets ◽  
The Other ◽  
Locally Convex Spaces ◽  
List Type ◽  
Topological Linear Space ◽  
Normed Spaces ◽  
Infinite Dimensional ◽  
Topological Linear ◽  
Locally Convex

Abstract By a tiling of a topological linear space X, we mean a covering of X by at least two closed convex sets, called tiles, whose nonempty interiors are pairwise disjoint. Study of tilings of infinite dimensional spaceswas initiated in the 1980's with pioneer papers by V. Klee. We prove some general properties of tilings of locally convex spaces, and then apply these results to study the existence of tilings of normed and Banach spaces by tiles possessing certain smoothness or rotundity properties. For a Banach space X, our main results are the following. (i) X admits no tiling by Fréchet smooth bounded tiles. (ii) If X is locally uniformly rotund (LUR), it does not admit any tiling by balls. (iii) On the other hand, some spaces, г uncountable, do admit a tiling by pairwise disjoint LUR bounded tiles.

Sign in / Sign up

Export Citation Format

Share Document