Entropy Optimization Problems for Modified Verma Measures in Primal and Dual Spaces

2021 ◽  
pp. 60-68
Author(s):  
Rohit Kumar Verma
Keyword(s):  
Optimization Problems ◽  
Dual Spaces ◽  
Entropy Optimization ◽  
Primal And Dual

scholarly journals Decentralized and parallel primal and dual accelerated methods for stochastic convex programming problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Darina Dvinskikh ◽  
Alexander Gasnikov
Keyword(s):  
Convex Optimization ◽  
Inverse Problems ◽  
Convex Programming ◽  
Data Science ◽  
Optimization Problems ◽  
Stochastic Gradient ◽  
Logarithmic Factor ◽  
Accelerated Methods ◽  
Convex Optimization Problems ◽  
Primal And Dual

Abstract We introduce primal and dual stochastic gradient oracle methods for decentralized convex optimization problems. Both for primal and dual oracles, the proposed methods are optimal in terms of the number of communication steps. However, for all classes of the objective, the optimality in terms of the number of oracle calls per node takes place only up to a logarithmic factor and the notion of smoothness. By using mini-batching technique, we show that the proposed methods with stochastic oracle can be additionally parallelized at each node. The considered algorithms can be applied to many data science problems and inverse problems.

scholarly journals Notes on Lipschitz Properties of Nonlinear Scalarization Functions with Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Fang Lu ◽  
Chun-Rong Chen
Keyword(s):  
Lipschitz Constant ◽  
Sufficient Conditions ◽  
Normed Spaces ◽  
Lipschitz Property ◽  
Nonlinear Scalarization ◽  
Dual Spaces ◽  
Lipschitz Constants ◽  
Primal And Dual ◽  
Vector Equilibrium ◽  
The One

Various kinds of nonlinear scalarization functions play important roles in vector optimization. Among them, the one commonly known as the Gerstewitz function is good at scalarizing. In linear normed spaces, the globally Lipschitz property of such function is deduced via primal and dual spaces approaches, respectively. The equivalence of both expressions for globally Lipschitz constants obtained by primal and dual spaces approaches is established. In particular, when the ordering cone is polyhedral, the expression for calculating Lipschitz constant is given. As direct applications of the Lipschitz property, several sufficient conditions for Hölder continuity of both single-valued and set-valued solution mappings to parametric vector equilibrium problems are obtained using the nonlinear scalarization approach.

scholarly journals GeodesicB-Preinvex Functions and Multiobjective Optimization Problems on Riemannian Manifolds

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Sheng-lan Chen ◽  
Nan-Jing Huang ◽  
Donal O'Regan
Keyword(s):  
Multiobjective Optimization ◽  
Riemannian Manifolds ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Properly Efficient Solution ◽  
Invex Functions ◽  
Multiobjective Optimization Problems ◽  
Dual Programming ◽  
Preinvex Functions ◽  
Primal And Dual

We introduce a class of functions called geodesicB-preinvex and geodesicB-invex functions on Riemannian manifolds and generalize the notions to the so-called geodesic quasi/pseudoB-preinvex and geodesic quasi/pseudoB-invex functions. We discuss the links among these functions under appropriate conditions and obtain results concerning extremum points of a nonsmooth geodesicB-preinvex function by using the proximal subdifferential. Moreover, we study a differentiable multiobjective optimization problem involving new classes of generalized geodesicB-invex functions and derive Kuhn-Tucker-type sufficient conditions for a feasible point to be an efficient or properly efficient solution. Finally, a Mond-Weir type duality is formulated and some duality results are given for the pair of primal and dual programming.

scholarly journals The Use of the Duality Principle to Solve Optimization Problems

2018 ◽  
Vol 6 (1) ◽  
pp. 33
Author(s):  
Rowland Jerry Okechukwu Ekeocha ◽  
Chukwunedum Uzor ◽  
Clement Anetor
Keyword(s):  
Optimization Problem ◽  
Dual Problem ◽  
Optimization Problems ◽  
Linear Program ◽  
Duality Gap ◽  
Duality Principle ◽  
Primal Problem ◽  
Optimal Values ◽  
Primal And Dual Problems ◽  
Primal And Dual

<p><span>The duality principle provides that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem. However the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition.<span>  </span>In other words given any linear program, there is another related linear program called the dual. In this paper, an understanding of the dual linear program will be developed. This understanding will give important insights into the algorithm and solution of optimization problem in linear programming. <span> </span>Thus the main concepts of duality will be explored by the solution of simple optimization problem.</span></p>

scholarly journals Slice convergence of parametrised sums of convex functions in non-reflexive spaces

1999 ◽  
Vol 60 (3) ◽  
pp. 429-458 ◽  
Author(s):  
Robert Wenczel ◽  
Andrew Eberhard
Keyword(s):  
Convex Functions ◽  
Mosco Convergence ◽  
Normed Spaces ◽  
Unified Framework ◽  
Dual Spaces ◽  
Slice Convergence ◽  
Sequential Continuity ◽  
Kuratowski Convergence ◽  
Primal And Dual ◽  
Fenchel Conjugation

The objectives of this study of slice convergence are two-fold. The first is to derive results regarding the passage of certain semi–convergences through Young–Fenchel conjugation. These semi–convergences arise from the splitting of the usual slice topology in the primal and dual spaces into (non-Hausdorff) topologies: the upper slice topology ; a topology generating a convergence closely resembling the bounded–weak* upper Kuratowski convergence; along with the respective primal and dual lower Kuratowski topologies. This gives rise to topological convergences not reliant on sequentially–based definitions found in many such studies, and associated topological continuity results for conjugation (in normed spaces), in contrast to the usual sequential continuity exhibited by analogues of Mosco convergence. The second objective is to study the passage of slice convergence through addition. Such sum theorems have been derived in other works and we establish previous theorems from a unified framework as well as obtaining a new result.

scholarly journals Proper efficiency and duality for vector valued optimization problems

1987 ◽  
Vol 43 (1) ◽  
pp. 21-34 ◽  
Author(s):  
T. Weir ◽  
B. Mond
Keyword(s):  
Convex Programming ◽  
Optimization Problems ◽  
Nonconvex Programming ◽  
Proper Efficiency ◽  
Weak Duality ◽  
Objective Functions ◽  
Duality Results ◽  
Type Relation ◽  
Primal And Dual ◽  
Vector Valued

AbstractThe duality results of Wolfe for scalar convex programming problems and some of the more recent duality results for scalar nonconvex programming problems are extended to vector valued programs. Weak duality is established using a ‘Pareto’ type relation between the primal and dual objective functions.

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