scholarly journals Characterizations of Asymptotic Cone of the Solution Set of a Composite Convex Optimization Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Zhe Chen
Keyword(s):  
Convex Optimization ◽  
Optimization Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Solution Set ◽  
Asymptotic Cone ◽  
Convex Optimization Problem ◽  
Composite Optimization ◽  
Convex Composite Optimization ◽  
Necessary And Sufficient

We characterize the asymptotic cone of the solution set of a convex composite optimization problem. We then apply the obtained results to study the necessary and sufficient conditions for the nonemptiness and compactness of the solution set of the problem. Our results generalize and improve some known results in literature.

A NEW KIND OF FUZZY RELATIONAL EQUATIONS

2010 ◽  
Vol 18 (03) ◽  
pp. 333-342 ◽  
Author(s):  
FENG QIN ◽  
PING FANG
Keyword(s):  
Unique Solvability ◽  
Complete Solution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Solution Set ◽  
Fuzzy Relational Equations ◽  
Necessary And Sufficient

In this paper, a new kind of fuzzy relational equations (FREs for short) A ∘R*x = b is first introduced, and then the problem of solving solution to the FREs is discussed, where A is an m × n matrix, x and b are an n and an m dimensional column vectors, respectively. More specifically, their solvability and unique solvability are investigated, the corresponding necessary and sufficient conditions are presented, the complete solution set is obtained. It is worth noting the method to construct the complete solution set.

scholarly journals Linear-quadratic optimization and some general hypotheses on optimal control

1999 ◽  
Vol 5 (4) ◽  
pp. 275-289 ◽  
Author(s):  
L. I. Rozonoer
Keyword(s):  
Optimal Control ◽  
Initial Data ◽  
Optimization Problem ◽  
Sufficient Conditions ◽  
Quadratic Optimization ◽  
Necessary And Sufficient Conditions ◽  
Maximum Condition ◽  
Linear Quadratic ◽  
Existence Of Optimal Control ◽  
Necessary And Sufficient

Necessary and sufficient conditions for existence of optimal control for all initial data are proved forLQ-optimization problem. If these conditions are fulfilled, necessary and sufficient conditions of optimality are formulated. Basing on the results, some general hypotheses on optimal control in terms of Pontryagin's maximum condition and Bellman's equation are proposed.

Nonlinear and linear entanglement witnesses for bipartite systems via exact convex optimization

2010 ◽  
Vol 10 (7&8) ◽  
pp. 562-579
Author(s):  
M. Jafarizadeh ◽  
A. Heshmati ◽  
K. Aghayar
Keyword(s):  
Convex Optimization ◽  
Single Particle ◽  
Sufficient Conditions ◽  
Quantum Systems ◽  
Necessary And Sufficient Conditions ◽  
Feasible Region ◽  
Entanglement Witnesses ◽  
Bipartite Quantum Systems ◽  
Linear And Nonlinear ◽  
Necessary And Sufficient

Linear and nonlinear entanglement witnesses for a given bipartite quantum systems are constructed. Using single particle feasible region, a way of constructing effective entanglement witnesses for bipartite systems is provided by exact convex optimization. Examples for some well known two qutrit quantum systems show these entanglement witnesses in most cases, provide necessary and sufficient conditions for separability of given bipartite system. Also this method is applied to a class of bipartite qudit quantum systems with details for d=3, 4 and 5.

scholarly journals A Note on Optimality Conditions for DC Programs Involving Composite Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Xiang-Kai Sun ◽  
Hong-Yong Fu
Keyword(s):  
Convex Optimization ◽  
Linear Operator ◽  
Convex Function ◽  
Optimality Conditions ◽  
Optimization Problem ◽  
Dc Programming ◽  
Sufficient Optimality Conditions ◽  
Convex Optimization Problem ◽  
Composite Function ◽  
Necessary And Sufficient

By using the formula of theε-subdifferential for the sum of a convex function with a composition of convex functions, some necessary and sufficient optimality conditions for a DC programming problem involving a composite function are obtained. As applications, a composed convex optimization problem, a DC optimization problem, and a convex optimization problem with a linear operator are examined at the end of this paper.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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