Criticality of Lagrange multipliers in extended nonlinear optimization
TL;DR: A systematic study of critical and noncritical multipliers in a general variational setting that covers, in particular, KKT systems in ENLP with establishing their verifiable characterizations as well as relationships between noncriticality and other stability notions in variational analysis.
Abstract: The paper is devoted to the study and applications of criticality of Lagrange multipliers in variational systems, which are associated with the class of problems in composite optimization known as ...
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TL;DR: In this article, a comprehensive study of composite models in variational analysis and optimization is presented, with the main attention paid to the new and rather large class of fully subamenable compositions, and the underlying theme of the study is a systematical replacement of conventional metric regularity and related requirements by much weaker metric subregulatity ones.
Abstract: The paper is devoted to a comprehensive study of composite models in variational analysis and optimization the importance of which for numerous theoretical, algorithmic, and applied issues of operations research is difficult to overstate. The underlying theme of our study is a systematical replacement of conventional metric regularity and related requirements by much weaker metric subregulatity ones that lead us to significantly stronger and completely new results of first-order and second-order variational analysis and optimization. In this way we develop extended calculus rules for first-order and second-order generalized differential constructions with paying the main attention in second-order variational theory to the new and rather large class of fully subamenable compositions. Applications to optimization include deriving enhanced no-gap second order optimality conditions in constrained composite models, complete characterizations of the uniqueness of Lagrange multipliers and strong metric subregularity of KKT systems in parametric optimization, etc.
24 citations
TL;DR: Wang et al. as mentioned in this paper applied machine learning models, including logistic regression (LR), the random forest model (RF), and the support vector machine (SVM) model, to assess landslide susceptibility in the Yangtze River's Three Gorges Reservoir region to analyze landslide events in the whole study region.
Abstract: The Three Gorges Reservoir region in China is the Yangtze River Economic Zone’s natural treasure trove. Its natural environment has an important role in development. The unique and fragile ecosystem in the Yangtze River’s Three Gorges Reservoir region is prone to natural disasters, including soil erosion, landslides, debris flows, landslides, and earthquakes. Therefore, to better alleviate these threats, an accurate and comprehensive assessment of the susceptibility of this area is required. In this study, based on the collection of relevant data and existing research results, we applied machine learning models, including logistic regression (LR), the random forest model (RF), and the support vector machine (SVM) model, to analyze landslide susceptibility in the Yangtze River’s Three Gorges Reservoir region to analyze landslide events in the whole study region. The models identified five categories (i.e., topographic, geological, ecological, meteorological, and human engineering activities), with nine independent variables, influencing landslide susceptibility. The accuracy of landslide susceptibility derived from different models and raster cells was then verified by the accuracy, recall, F1-score, ROC curve, and AUC of each model. The results illustrate that the accuracy of different machine learning algorithms is ranked as SVM > RF > LR. The LR model has the lowest generalization ability. The SVM model performs well in all regions of the study area, with an AUC value of 0.9708 for the entire Three Gorges Reservoir area, indicating that the SVM model possesses a strong spatial generalization ability as well as the highest robustness and can be adapted as a real-time model for assessing regional landslide susceptibility.
10 citations
TL;DR: In this paper, the primal superlinear convergence of the quasi-Newton sequential quadratic programming (SQP) method for piecewise linear-quadratic composite optimization problems was analyzed under the second-order sufficient condition and the uniqueness of Lagrange multipliers.
Abstract: This paper mainly concerns with the primal superlinear convergence of the quasi-Newton sequential quadratic programming (SQP) method for piecewise linear-quadratic composite optimization problems We show that the latter primal superlinear convergence can be justified under the noncriticality of Lagrange multipliers and a version of the Dennis-More condition Furthermore, we show that if we replace the noncriticality condition with the second-order sufficient condition, this primal superlinear convergence is equivalent with an appropriate version of the Dennis-More condition We also recover Bonnans’ result in (Appl Math Optim 29, 161–186, 1994) for the primal-dual superlinear of the basic SQP method for this class of composite problems under the second-order sufficient condition and the uniqueness of Lagrange multipliers To achieve these goals, we first obtain an extension of the reduction lemma for convex Piecewise linear-quadratic functions and then provide a comprehensive analysis of the noncriticality of Lagrange multipliers for composite problems We also establish certain primal estimates for KKT systems of composite problems, which play a significant role in our local convergence analysis of the quasi-Newton SQP method
4 citations
TL;DR: It is a well-known phenomenon that the presence of critical Lagrange multipliers in constrained optimization problems may cause a deterioration of the convergence speed of primal-dual Newton-type mappings.
Abstract: It is a well-known phenomenon that the presence of critical Lagrange multipliers in constrained optimization problems may cause a deterioration of the convergence speed of primal-dual Newton-type m...
2 citations
TL;DR: In this article, the second-order epi-differentiability of the indicator function has been studied for disjunctive constrained problems, including finite union of parabolically derivable and regular sets.
Abstract: In this paper, we examine the properly twice epi-differentiability and compute the second order epi-subderivative of the indicator function to a class of sets including the finite union of parabolically derivable and parabolically regular sets. In this way, we provide no-gap second order optimality conditions for a disjunctive constrained problem. Moreover, we derive applications of our results to some types of disjunctive programs.
1 citations
References
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01 Jan 2006
1,528 citations
TL;DR: In this article, the authors consider some basic properties of nonsmooth and set-valued mappings (multifunctions) connected with open and inverse mapping principles, distance estimates to the level sets (metric regularity), and a locally Lipschitzian behavior.
Abstract: We consider some basic properties of nonsmooth and set-valued mappings (multifunctions) connected with open and inverse mapping principles, distance estimates to the level sets (metric regularity), and a locally Lipschitzian behavior. These properties have many important applications to various problems in nonlinear analysis, optimization, control theory, etc., especially for studying sensitivity and stability questions with respect to perturbations of initial data and parameters. We establish interrelations between these properties and prove effective criteria for their fulfillment stated in terms of robust generalized derivatives for multifunctions and nonsmooth mappings
345 citations
TL;DR: Property of prox-regularity of the essential objective function and positive definiteness of its coderivative Hessian are the keys to the Lipschitzian stability of local solutions to finite-dimensional parameterized optimization problems in a very general setting.
Abstract: Necessary and sufficient conditions are obtained for the Lipschitzian stability of local solutions to finite-dimensional parameterized optimization problems in a very general setting. Properties of prox-regularity of the essential objective function and positive definiteness of its coderivative Hessian are the keys to these results. A previous characterization of tilt stability arises as a special case.
126 citations
"Criticality of Lagrange multipliers..." refers background in this paper
...nsure that critical multipliers corresponding to this minimizer do not arise. It is conjectured in [10], based on preliminary results for NLPs, that full stability of local minimizers in the sense of [7] rules out the appearance of critical multiplies. This conjecture was verified in [15] for polyhedral problems of type (1.1) with convex piecewise linear functions θ. Now we justify this conjecture in ...
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...ocal minimizers allows us to rule out the appearance of critical multipliers associated with such local optimal solutions. It is conjectured in [10] that fully stable local minimizers in the sense of [7] are appropriate candidate for excluding critical multipliers. This conjecture is affirmatively verified in [14] for problems (1.1) with θ = θY,B where B = 0. Now we are able to extend this result to the...
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TL;DR: Based on second-order generalized differential tools of variational anal- ysis, necessary and sufficient conditions are obtained for fully stable local minimizers in general classes of constrained optimization problems, including problems of composite optimization, mathemati- cal programs with polyhedral constraints, as well as problems of extended and classical nonlinear programming with twice continuously differentiable data.
Abstract: This paper is mainly devoted to the study of the so-called full Lipschitzian stability of local solutions to finite-dimensional parameterized problems of constrained optimization, which has been well recognized as a very important property from the viewpoints of both optimization theory and its applications. Based on second-order generalized differential tools of variational anal- ysis, we obtain necessary and sufficient conditions for fully stable local minimizers in general classes of constrained optimization problems, including problems of composite optimization, mathemati- cal programs with polyhedral constraints, as well as problems of extended and classical nonlinear programming with twice continuously differentiable data.
73 citations
"Criticality of Lagrange multipliers..." refers background in this paper
...This conjecture is affirmatively verified in [14] for problems (1....
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TL;DR: In this paper, the authors studied the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) solution mapping for a large class of interesting conic programming problems at a locally optimal solution.
Abstract: This paper is devoted to studying the robust isolated calmness of the Karush--Kuhn--Tucker (KKT) solution mapping for a large class of interesting conic programming problems (including most commonly known ones arising from applications) at a locally optimal solution. Under the Robinson constraint qualification, we show that the KKT solution mapping is robustly isolated calm if and only if both the strict Robinson constraint qualification and the second order sufficient condition hold. This implies, among others, that at a locally optimal solution the second order sufficient condition is needed for the KKT solution mapping to have the Aubin property.
41 citations