scholarly journals Note on the topological degree of the subdifferential of a lower semi-continuous convex function

1998 ◽  
Vol 126 (10) ◽  
pp. 2905-2908 ◽  
Author(s):  
Sergiu Aizicovici ◽  
Yuqing Chen
Keyword(s):  
Convex Function ◽  
Topological Degree ◽  
Lower Semi ◽  
Continuous Convex Function

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 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 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.

Some remarks on probability inequalities for sums of bounded convex random variables

1975 ◽  
Vol 12 (1) ◽  
pp. 155-158 ◽  
Author(s):  
M. Goldstein
Keyword(s):  
Convex Function ◽  
Exponential Convergence ◽  
Random Variables ◽  
Upper Bounds ◽  
Independent Random Variables ◽  
Probability Inequalities ◽  
Continuous Convex Function ◽  
Bounded Convex

Let X1, X2, · ··, Xn be independent random variables such that ai ≦ Xi ≦ bi, i = 1,2,…n. A class of upper bounds on the probability P(S−ES ≧ nδ) is derived where S = Σf(Xi), δ > 0 and f is a continuous convex function. Conditions for the exponential convergence of the bounds are discussed.

scholarly journals Multi-valued backward stochastic differential equations with regime switching

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ruijuan Deng ◽  
Yong Ren
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Regime Switching ◽  
Original System ◽  
Backward Stochastic Differential Equations ◽  
Lower Semi ◽  
Finite State ◽  
Uniqueness Of The Solution ◽  
Continuous Convex Function

AbstractThe paper considers a class of multi-valued backward stochastic differential equations with subdifferential of a lower semi-continuous convex function with regime switching, whose generator is a continuous-time Markov chain with a finite state space. Firstly, we get the existence and uniqueness of the solution by the penalization method. Secondly, we prove that the solution of the original system is weakly convergent. Finally, we give an application to the homogenization of a class of multi-valued PDEs with Markov chain.

scholarly journals Another Aspect of Triangle Inequality

2011 ◽  
Vol 2011 ◽  
pp. 1-5
Author(s):  
Kichi-Suke Saito ◽  
Runling An ◽  
Hiroyasu Mizuguchi ◽  
Ken-Ichi Mitani
Keyword(s):  
Convex Function ◽  
Triangle Inequality ◽  
Unit Interval ◽  
Usual Sense ◽  
Continuous Convex Function

We introduce the notion of ψ-norm by considering the fact that an absolute normalized norm on C2 corresponds to a continuous convex function ψ on the unit interval [0,1] with some conditions. This is a generalization of the notion of q-norm introduced by Belbachir et al. (2006). Then we show that a ψ-norm is a norm in the usual sense.

scholarly journals Hausdorff dimension of the nondifferentiability set of a convex function

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5827-5831
Author(s):  
Reza Mirzaie
Keyword(s):  
Riemannian Manifold ◽  
Hausdorff Dimension ◽  
Convex Function ◽  
Open Subset ◽  
Upper Bound ◽  
Continuous Convex Function

We find an upper bound for the Hausdorff dimension of the nondifferentiability set of a continuous convex function defined on a Riemannian manifold. As an application, we show that the boundary of a convex open subset of Rn, n ? 2, has Hausdorff dimension at most n - 2.

scholarly journals REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY A LÉVY PROCESS

2009 ◽  
Vol 50 (4) ◽  
pp. 486-500 ◽  
Author(s):  
YONG REN ◽  
XILIANG FAN
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lévy Process ◽  
Backward Stochastic Differential Equations ◽  
Levy Process ◽  
Penalization Method ◽  
Lower Semi ◽  
Uniqueness Of The Solution ◽  
Teugels Martingales ◽  
Continuous Convex Function

AbstractIn this paper, we deal with a class of reflected backward stochastic differential equations (RBSDEs) corresponding to the subdifferential operator of a lower semi-continuous convex function, driven by Teugels martingales associated with a Lévy process. We show the existence and uniqueness of the solution for RBSDEs by means of the penalization method. As an application, we give a probabilistic interpretation for the solutions of a class of partial differential-integral inclusions.

scholarly journals CONVEX MULTIVARIABLE TRACE FUNCTIONS

2002 ◽  
Vol 14 (07n08) ◽  
pp. 631-648 ◽  
Author(s):  
ELLIOTT H. LIEB ◽  
GERT K. PEDERSEN
Keyword(s):  
Convex Function ◽  
Short Proof ◽  
Trace Function ◽  
C Algebra ◽  
Operator Mean ◽  
Lower Semi ◽  
Continuous Trace ◽  
Densely Defined

For any densely defined, lower semi-continuous trace τ on a C*-algebra A with mutually commuting C*-subalgebras A1, A2, … An, and a convex function f of n variables, we give a short proof of the fact that the function (x1, x2, …, xn)→ τ (f (x1, x2, …, xn)) is convex on the space [Formula: see text]. If furthermore the function f is log-convex or root-convex, so is the corresponding trace function. We also introduce a generalization of log-convexity and root-convexity called ℓ-convexity, show how it applies to traces, and give some examples. In particular we show that the Kadison–Fuglede determinant is concave and that the trace of an operator mean is always dominated by the corresponding mean of the trace values.

Sign in / Sign up

Export Citation Format

Share Document