First-Order Algorithms for Optimization Problems with a Maximum Eigenvalue/Singular Value Cost and or Constraints

2001 ◽  
pp. 197-220
Author(s):  
Elijah Polak
Keyword(s):  
Optimization Problems ◽  
Singular Value ◽  
Maximum Eigenvalue ◽  
First Order

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.

scholarly journals A decoupling algorithm with first-order asymptotic integration for reliability-based design optimization

2018 ◽  
Vol 10 (9) ◽  
pp. 168781401879333 ◽  
Author(s):  
Zhiliang Huang ◽  
Tongguang Yang ◽  
Fangyi Li
Keyword(s):  
Design Optimization ◽  
Reliability Analysis ◽  
Optimization Problems ◽  
Asymptotic Integration ◽  
Probabilistic Constraints ◽  
Reliability Based Design Optimization ◽  
First Order Reliability Method ◽  
First Order ◽  
Reliability Based Design ◽  
Reliability Method

Conventional decoupling approaches usually employ first-order reliability method to deal with probabilistic constraints in a reliability-based design optimization problem. In first-order reliability method, constraint functions are transformed into a standard normal space. Extra non-linearity introduced by the non-normal-to-normal transformation may increase the error in reliability analysis and then result in the reliability-based design optimization analysis with insufficient accuracy. In this article, a decoupling approach is proposed to provide an alternative tool for the reliability-based design optimization problems. To improve accuracy, the reliability analysis is performed by first-order asymptotic integration method without any extra non-linearity transformation. To achieve high efficiency, an approximate technique of reliability analysis is given to avoid calculating time-consuming performance function. Two numerical examples and an application of practical laptop structural design are presented to validate the effectiveness of the proposed approach.

scholarly journals Sensitivity analysis of Wasserstein distributionally robust optimization problems

2021 ◽  
Vol 477 (2256) ◽  
Author(s):  
Daniel Bartl ◽  
Samuel Drapeau ◽  
Jan Obłój ◽  
Johannes Wiesel
Keyword(s):  
Model Uncertainty ◽  
Optimization Problems ◽  
Order Approximation ◽  
Mathematical Finance ◽  
Ordinary Least Squares ◽  
Linear Constraints ◽  
Least Squares Regression ◽  
Lasso Regression ◽  
First Order ◽  
Non Parametric

We consider sensitivity of a generic stochastic optimization problem to model uncertainty. We take a non-parametric approach and capture model uncertainty using Wasserstein balls around the postulated model. We provide explicit formulae for the first-order correction to both the value function and the optimizer and further extend our results to optimization under linear constraints. We present applications to statistics, machine learning, mathematical finance and uncertainty quantification. In particular, we provide an explicit first-order approximation for square-root LASSO regression coefficients and deduce coefficient shrinkage compared to the ordinary least-squares regression. We consider robustness of call option pricing and deduce a new Black–Scholes sensitivity, a non-parametric version of the so-called Vega. We also compute sensitivities of optimized certainty equivalents in finance and propose measures to quantify robustness of neural networks to adversarial examples.

scholarly journals On the Convergence of (Stochastic) Gradient Descent with Extrapolation for Non-Convex Minimization

2019 ◽  
Author(s):  
Yi Xu ◽  
Zhuoning Yuan ◽  
Sen Yang ◽  
Rong Jin ◽  
Tianbao Yang
Keyword(s):  
Convex Optimization ◽  
Gradient Descent ◽  
Optimization Problems ◽  
Upper Bounds ◽  
Convex Minimization ◽  
Stochastic Gradient ◽  
Stochastic Gradient Descent ◽  
First Order ◽  
Convex Optimization Problems ◽  
Learning Tasks

Extrapolation is a well-known technique for solving convex optimization and variational inequalities and recently attracts some attention for non-convex optimization. Several recent works have empirically shown its success in some machine learning tasks. However, it has not been analyzed for non-convex minimization and there still remains a gap between the theory and the practice. In this paper, we analyze gradient descent  and stochastic gradient descent with extrapolation for finding an approximate first-order stationary point in smooth non-convex optimization problems. Our convergence upper bounds show that the algorithms with extrapolation can be accelerated than without extrapolation.

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