M. Ebrahim Sarabi

Bio: M. Ebrahim Sarabi is an academic researcher from Miami University. The author has contributed to research in topics: Variational analysis & Lagrange multiplier. The author has an hindex of 9, co-authored 28 publications receiving 263 citations. Previous affiliations of M. Ebrahim Sarabi include Wayne State University.

Full Stability of Locally Optimal Solutions in Second-Order Cone Programs

Abstract: The paper presents complete characterizations of Lipschitzian full stability of locally optimal solutions to second-order cone programs (SOCPs) expressed entirely in terms of their initial data. These characterizations are obtained via appropriate versions of the quadratic growth and strong second-order sufficient conditions under the corresponding constraint qualifications. We also establish close relationships between full stability of local minimizers for SOCPs and strong regularity of the associated generalized equations at nondegenerate points. Our approach is mainly based on advanced tools of second-order variational analysis and generalized differentiation.

Critical multipliers in variational systems via second-order generalized differentiation

Abstract: In this paper we introduce the notions of critical and noncritical multipliers for variational systems and extend to a general framework the corresponding notions by Izmailov and Solodov developed for classical Karush–Kuhn–Tucker (KKT) systems. It has been well recognized that critical multipliers are largely responsible for slow convergence of major primal–dual algorithms of optimization. The approach of this paper allows us to cover KKT systems arising in various classes of smooth and nonsmooth problems of constrained optimization including composite optimization, minimax problems, etc. Concentrating on a polyhedral subdifferential case and employing recent results of second-order subdifferential theory, we obtain complete characterizations of critical and noncritical multipliers via the problem data. It is shown that noncriticality is equivalent to a certain calmness property of a perturbed variational system and that critical multipliers can be ruled out by full stability of local minimizers in problems of composite optimization. For the latter class we establish the equivalence between noncriticality of multipliers and robust isolated calmness of the associated solution map and then derive explicit characterizations of these notions via appropriate second-order sufficient conditions. It is finally proved that the Lipschitz-like/Aubin property of solution maps yields their robust isolated calmness.

Twice Epi-Differentiability of Extended-Real-Valued Functions with Applications in Composite Optimization

Abstract: The paper is devoted to the study of the twice epi-differentiablity of extended-real-valued functions, with an emphasis on functions satisfying a certain composite representation. This will be cond...

Parabolic Regularity in Geometric Variational Analysis

Abstract: The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in variational analysis for more than two decades while being largely underinvestigated. We discover here that parabolic regularity is the key to derive new calculus rules and computation formulas for major second-order generalized differential constructions of variational analysis in connection with some properties of sets that go back to classical differential geometry and geometric measure theory. The established results of second-order variational analysis and generalized differentiation, being married to the developed calculus of parabolic regularity, allow us to obtain novel applications to both qualitative and quantitative/numerical aspects of constrained optimization including second-order optimality conditions, augmented Lagrangians, etc. under weak constraint qualifications.

Variational Analysis of Composite Models with Applications to Continuous Optimization

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.

Perturbation Analysis Of Optimization Problems

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New fractional error bounds for polynomial systems with applications to Hölderian stability in optimization and spectral theory of tensors

Abstract: In this paper we derive new fractional error bounds for polynomial systems with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved polynomials. Our major result extends the existing error bounds from the system involving only a single polynomial to a general polynomial system and do not require any regularity assumptions. In this way we resolve, in particular, some open questions posed in the literature. The developed techniques are largely based on variational analysis and generalized differentiation, which allow us to establish, e.g., a nonsmooth extension of the seminal ?ojasiewicz's gradient inequality to maxima of polynomials with explicitly determined exponents. Our major applications concern quantitative Holderian stability of solution maps for parameterized polynomial optimization problems and nonlinear complementarity systems with polynomial data as well as high-order semismooth properties of the eigenvalues of symmetric tensors.

Complete Characterizations of Tilt Stability in Nonlinear Programming under Weakest Qualification Conditions

Abstract: This paper is devoted to the study of tilt stability of local minimizers for classical nonlinear programs with equality and inequality constraints in finite dimensions described by twice continuous...

Gilbert综合征二例

Abstract: Gilbert综合征(Gilbert syndrome,GS)又称先天性非溶血性黄疸,是以遗传性慢性轻度非结合型高胆红素血症,或同时伴有肝脏色素清除功能障碍,但无肝脏结构改变或肝脏其他功能障碍为特点的一类罕见肝病.2008年1月至2009年2月河北省承德县医院和河北医科大学第三医院收治2例GS患者,均经病理证实,现报道如下。