Primal and Dual Variables for New QP-free Method without a Penalty Function and a Filter

2012 ◽  
Author(s):  
Dingguo Pu ◽  
Youlin Shang
Keyword(s):  
Penalty Function ◽  
Dual Variables ◽  
Primal And Dual

scholarly journals Improved Lagrangian-PPA based prediction correction method for linearly constrained convex optimization

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yanfei You ◽  
Suhong Jiang
Keyword(s):  
Convex Optimization ◽  
Optimization Problem ◽  
Correction Method ◽  
Lagrangian Function ◽  
Convex Optimization Problem ◽  
Constrained Convex Optimization ◽  
Linearly Constrained ◽  
Dual Variables ◽  
Primal And Dual ◽  
Linearly Constrained Convex Optimization

<p style='text-indent:20px;'>This paper presents an improved Lagrangian-PPA based prediction correction method to solve linearly constrained convex optimization problem. At each iteration, the predictor is achieved by minimizing the proximal Lagrangian function with respect to the primal and dual variables. These optimization subproblems involved either admit analytical solutions or can be solved by a fast algorithm. The new update is generated by using the information of the current iterate and the predictor, as well as an appropriately chosen stepsize. Compared with the existing PPA based method, the parameters are relaxed. We also establish the convergence and convergence rate of the proposed method. Finally, numerical experiments are conducted to show the efficiency of our Lagrangian-PPA based prediction correction method.</p>

QP-Free Method for Nonlinear Programming Problems

2011 ◽  
Vol 467-469 ◽  
pp. 882-887
Author(s):  
Ai Ping Jiang ◽  
Feng Wen Huang
Keyword(s):  
Linear Independence ◽  
Accumulation Point ◽  
Strict Complementarity ◽  
First Order ◽  
Ncp Function ◽  
Complementarity Condition ◽  
Dual Variables ◽  
Primal And Dual ◽  
Weaker Conditions ◽  
Quasi Newton

In this paper, A QP-free feasible method was proposed to obtain the local convergence under some weaker conditions for the minimization of a smooth function subject to smooth inequalities. Based on the solutions of linear systems of equation reformulation of the KKT optimality conditions, this method uses the 3-1 NCP function[1].The method is iterative, which means each iteration can be viewed as a perturbation of a Newton or Quasi Newton on both the primal and dual variables for the solution of the equalities in the KKT first order conditions of optimality, and the feasibility of all iterations is ensured in this method. In particular, this method is implementable and globally convergent without assuming the strict complementarity condition, the isolation of the accumulation point and the linear independence of the gradients of active constrained functions. The method has also superlinear convergence rate under some mild conditions.

scholarly journals Dual Modelling of Permutation and Injection Problems

2004 ◽  
Vol 21 ◽  
pp. 357-391 ◽  
Author(s):  
B. Hnich ◽  
B. M. Smith ◽  
T. Walsh
Keyword(s):  
Dual Variable ◽  
Combined Model ◽  
Multiple Viewpoints ◽  
Constraint Program ◽  
Decision Variables ◽  
Primal Dual ◽  
Combined Models ◽  
Dual Variables ◽  
Primal And Dual ◽  
Permutation Problems

When writing a constraint program, we have to choose which variables should be the decision variables, and how to represent the constraints on these variables. In many cases, there is considerable choice for the decision variables. Consider, for example, permutation problems in which we have as many values as variables, and each variable takes an unique value. In such problems, we can choose between a primal and a dual viewpoint. In the dual viewpoint, each dual variable represents one of the primal values, whilst each dual value represents one of the primal variables. Alternatively, by means of channelling constraints to link the primal and dual variables, we can have a combined model with both sets of variables. In this paper, we perform an extensive theoretical and empirical study of such primal, dual and combined models for two classes of problems: permutation problems and injection problems. Our results show that it often be advantageous to use multiple viewpoints, and to have constraints which channel between them to maintain consistency. They also illustrate a general methodology for comparing different constraint models.

scholarly journals A QUASI-VARIATIONAL INEQUALITY PROBLEM IN SUPERCONDUCTIVITY

2010 ◽  
Vol 20 (05) ◽  
pp. 679-706 ◽  
Author(s):  
JOHN W. BARRETT ◽  
LEONID PRIGOZHIN
Keyword(s):  
Variational Inequality ◽  
Electric Fields ◽  
Variational Inequality Problem ◽  
Existence Of A Solution ◽  
Numerical Examples ◽  
Inequality Problem ◽  
Dual Formulation ◽  
Quasi Variational Inequality ◽  
Dual Variables ◽  
Primal And Dual

We derive a class of analytical solutions and a dual formulation of a scalar two-space-dimensional quasi-variational inequality problem in applied superconductivity. We approximate this formulation by a fully practical finite element method based on the lowest order Raviart–Thomas element, which yields approximations to both the primal and dual variables (the magnetic and electric fields). We prove the subsequence convergence of this approximation, and hence prove the existence of a solution to both the dual and primal formulations, for strictly star-shaped domains. The effectiveness of the approximation is illustrated by numerical examples with and without this domain restriction.

Rule-Based LTV and Penalty Function for Concentration Risk

2012 ◽  
Author(s):  
Yongwoong Lee ◽  
Yiran Zhang ◽  
Ser-Huang Poon
Keyword(s):  
Penalty Function ◽  
Rule Based

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.

Dynamic Buckling of Cylindrical Shells under Axial Impact in Hamiltonian System

2012 ◽  
Vol 13 (1) ◽  
Author(s):  
Jia-Bin Sun ◽  
Xin-Sheng Xu ◽  
Chee-Wah Lim
Keyword(s):  
Hamiltonian System ◽  
Critical Load ◽  
Impact Load ◽  
Dynamic Buckling ◽  
Inertia Force ◽  
Elastic Cylindrical Shell ◽  
Symplectic Method ◽  
Buckling Mode ◽  
Axial Impact ◽  
Dual Variables

AbstractIn this paper, the dynamic buckling of an elastic cylindrical shell subjected to an axial impact load is analyzed in Hamiltonian system. By employing a symplectic method, the traditional governing equations are transformed into Hamiltonian canonical equations in dual variables. In this system, the critical load and buckling mode are reduced to solving symplectic eigenvalues and eigensolutions respectively. The result shows that the critical load relates with boundary conditions, thickness of the shell and radial inertia force. And the corresponding buckling modes present some local shapes. Besides, the process of dynamic buckling is related to the stress wave, the critical load and buckling mode depend upon the impacted time. This paper gives analytically and numerically some new rules of the buckling problem, which is useful for designing shell structures.

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