scholarly journals Second-Order Multiplier Iteration Based on a Class of Nonlinear Lagrangians

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Yong-Hong Ren
Keyword(s):  
Nonlinear Programming ◽  
Optimization Problems ◽  
Inequality Constraints ◽  
Second Order ◽  
Constrained Optimization Problems ◽  
Lipschitz Continuous ◽  
First Order ◽  
Nonlinear Lagrangians ◽  
Nonlinear Lagrangian ◽  
Nonlinear Lagrangian Algorithm

Nonlinear Lagrangian algorithm plays an important role in solving constrained optimization problems. It is known that, under appropriate conditions, the sequence generated by the first-order multiplier iteration converges superlinearly. This paper aims at analyzing the second-order multiplier iteration based on a class of nonlinear Lagrangians for solving nonlinear programming problems with inequality constraints. It is suggested that the sequence generated by the second-order multiplier iteration converges superlinearly with order at least two if in addition the Hessians of functions involved in problem are Lipschitz continuous.

Numerical Techniques for Nonlinear Lagrangian Dynamical Systems and Their Application

2012 ◽  
Vol 476-478 ◽  
pp. 1513-1516
Author(s):  
Jin Li
Keyword(s):  
Dynamical Systems ◽  
General Framework ◽  
Optimization Problems ◽  
Lagrangian Function ◽  
Lagrangian System ◽  
Numerical Techniques ◽  
Constrained Optimization Problems ◽  
First Order ◽  
Nonlinear Lagrangian ◽  
Special Case

We provide a general framework for solving constrained optimization problems, this framework relies on dynamical systems using a class of nonlinear Lagrangian function, we construct a first order derivatives based and a second order derivatives based differential systems. Under this framework, We show that the exponential Lagrangian system as the special case is discussed.

The Rate of Convergence of a NLM Based on F–B NCP for Constrained Optimization Problems Without Strict Complementarity

2015 ◽  
Vol 32 (03) ◽  
pp. 1550012 ◽  
Author(s):  
Suxiang He ◽  
Liwei Zhang ◽  
Jie Zhang
Keyword(s):  
Constrained Optimization ◽  
Rate Of Convergence ◽  
Constraint Qualification ◽  
Optimization Problems ◽  
Augmented Lagrangian Method ◽  
Inequality Constraints ◽  
Constrained Optimization Problems ◽  
Strict Complementarity ◽  
Linear Rate ◽  
Nonlinear Lagrangian

It is well-known that the linear rate of convergence can be established for the classical augmented Lagrangian method for constrained optimization problems without strict complementarity. Whether this result is still valid for other nonlinear Lagrangian methods (NLM) is an interesting problem. This paper proposes a nonlinear Lagrangian function based on Fischer–Burmeister (F–B) nonlinear complimentarity problem (NCP) function for constrained optimization problems. The rate of convergence of this NLM is analyzed under the linear independent constraint qualification and the strong second-order sufficient condition without strict complementarity when subproblems are assumed to be solved exactly and inexactly, respectively. Interestingly, it is demonstrated that the Lagrange multipliers associating with inactive inequality constraints at the local minimum point converge to zeros superlinearly. Several illustrative examples are reported to show the behavior of the NLM.

scholarly journals Coupling Elephant Herding with Ordinal Optimization for Solving the Stochastic Inequality Constrained Optimization Problems

2020 ◽  
Vol 10 (6) ◽  
pp. 2075 ◽  
Author(s):  
Shih-Cheng Horng ◽  
Shieh-Shing Lin
Keyword(s):  
Optimization Problems ◽  
Solution Space ◽  
Inequality Constraints ◽  
Optimization Methods ◽  
Ordinal Optimization ◽  
Np Hard ◽  
Constrained Optimization Problems ◽  
Inequality Constrained Optimization ◽  
Hard Problems ◽  
Np Hard Problems

The stochastic inequality constrained optimization problems (SICOPs) consider the problems of optimizing an objective function involving stochastic inequality constraints. The SICOPs belong to a category of NP-hard problems in terms of computational complexity. The ordinal optimization (OO) method offers an efficient framework for solving NP-hard problems. Even though the OO method is helpful to solve NP-hard problems, the stochastic inequality constraints will drastically reduce the efficiency and competitiveness. In this paper, a heuristic method coupling elephant herding optimization (EHO) with ordinal optimization (OO), abbreviated as EHOO, is presented to solve the SICOPs with large solution space. The EHOO approach has three parts, which are metamodel construction, diversification and intensification. First, the regularized minimal-energy tensor-product splines is adopted as a metamodel to approximately evaluate fitness of a solution. Next, an improved elephant herding optimization is developed to find N significant solutions from the entire solution space. Finally, an accelerated optimal computing budget allocation is utilized to select a superb solution from the N significant solutions. The EHOO approach is tested on a one-period multi-skill call center for minimizing the staffing cost, which is formulated as a SICOP. Simulation results obtained by the EHOO are compared with three optimization methods. Experimental results demonstrate that the EHOO approach obtains a superb solution of higher quality as well as a higher computational efficiency than three optimization methods.

scholarly journals Relations between Abs-Normal NLPs and MPCCs. Part 2: Weak Constraint Qualifications

2021 ◽  
Vol Volume 2 (Original research articles>) ◽  
Author(s):  
Lisa C. Hegerhorst-Schultchen ◽  
Christian Kirches ◽  
Marc C. Steinbach
Keyword(s):  
Normal Form ◽  
Optimization Problems ◽  
Constraint Qualifications ◽  
Inequality Constraints ◽  
First Order ◽  
Nonlinear Programs ◽  
Mathematical Programs ◽  
Ongoing Effort ◽  
Equality And Inequality Constraints ◽  
Smooth Optimization

This work continues an ongoing effort to compare non-smooth optimization problems in abs-normal form to Mathematical Programs with Complementarity Constraints (MPCCs). We study general Nonlinear Programs with equality and inequality constraints in abs-normal form, so-called Abs-Normal NLPs, and their relation to equivalent MPCC reformulations. We introduce the concepts of Abadie's and Guignard's kink qualification and prove relations to MPCC-ACQ and MPCC-GCQ for the counterpart MPCC formulations. Due to non-uniqueness of a specific slack reformulation suggested in [10], the relations are non-trivial. It turns out that constraint qualifications of Abadie type are preserved. We also prove the weaker result that equivalence of Guginard's (and Abadie's) constraint qualifications for all branch problems hold, while the question of GCQ preservation remains open. Finally, we introduce M-stationarity and B-stationarity concepts for abs-normal NLPs and prove first order optimality conditions corresponding to MPCC counterpart formulations.

scholarly journals Sparse Constrained Reconstruction for Accelerating Parallel Imaging Based on Variable Splitting Method

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Wenlong Xu ◽  
Xiaofang Liu ◽  
Xia Li
Keyword(s):  
Magnetic Resonance ◽  
Parallel Imaging ◽  
Optimization Problems ◽  
Splitting Method ◽  
Second Order ◽  
Constrained Reconstruction ◽  
Reconstruction Process ◽  
First Order ◽  
Variable Splitting ◽  
Reconstruction Methods

Parallel imaging is a rapid magnetic resonance imaging technique. For the ill-conditioned problem, noise and aliasing artifacts are amplified during the reconstruction process and are serious especially for high accelerating imaging. In this paper, a sparse constrained reconstruction problem is proposed for parallel imaging, and an effective solution based on the variable splitting method is contrived. First-order and second-order norm optimization problems are first split, and then they are transferred to unconstrained minimization problem by the augmented Lagrangian method. At last, first-order norm and second-order norm optimization problems are alternatively resolved by different methods. With a discrepancy principle as the stopping criterion, analysis of simulated and actual parallel magnetic resonance image reconstruction is presented and discussed. Compared with the routine parallel imaging reconstruction methods, the results show that the noise and aliasing artifacts in the reconstructed image are evidently reduced at large acceleration factors.

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