First and second order analysis of nonlinear semidefinite programs
Mathematical ProgrammingPublished 1 April 1997
Alexander Shapiro
Citations193
SJR quartileQ1
SJR score1.73
SNIP2.20
Generate an AI Snapshot to get a quick, structured summary of this paper.
Study Snapshot
ObjectiveStudy objective
MethodsResearch methodology
PopulationPopulation studied
Sample sizeSample sizes
OutcomesStudy outcomes here
ResultsStudy results comes here
LimitationsResearch study limitations comes here
A concise AI-generated summary of the paper will appear here once you click Generate AI Snapshot.
TL;DR
Convexity, duality and first-order optimality conditions for nonlinear semidefinite programming problems are presented and sensitivity analysis of such programs is discussed.
Abstract
In this paper we study nonlinear semidefinite programming problems. Convexity, duality and first-order optimality conditions for such problems are presented. A second-order analysis is also given. Second-order necessary and sufficient optimality conditions are derived. Finally, sensitivity analysis of such programs is discussed.
Keywords
Computer ScienceMathematicsEngineering
Cambridge University Press eBooksTopics in Matrix Analysis
9,501 Citations1991Roger A. Horn, Charles R. Johnson
Journal of the American Statistical AssociationInequalities: Theory of Majorization and its Applications.
4,984 Citations1981Barry C. Arnold, Albert W. Marshall +1 more
Society for Industrial and Applied Mathematics eBooksInterior-Point Polynomial Algorithms in Convex Programming
4,275 Citations1994Yurii Nesterov, Arkadi Nemirovski
This book describes the first unified theory of polynomial-time interior-point methods, and describes several of the new algorithms described, e.g., the projective method, which have been implemented, tested on "real world" problems, and found to be extremely efficient in practice.
Graduate texts in mathematicsStable Mappings and Their Singularities
1,358 Citations1973Martin Golubitsky, Victor Guillemin
Society for Industrial and Applied Mathematics eBooks1. Conjugate Duality and Optimization
898 Citations1974R. T. Rockafellar
SIAM Journal on OptimizationPrimal-Dual Interior-Point Methods for Semidefinite Programming: Convergence Rates, Stability and Numerical Results
486 Citations1998Farid Alizadeh, Jean-Pierre A. Haeberly +1 more
The XZ+ZX method is more robust with respect to its ability to step close to the boundary, converges more rapidly, and achieves higher accuracy than other methods considered, including Mehrotra predictor-corrector variants and issues of numerical stability.
TechnometricsDiscrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method
395 Citations1995Jason H. Goodfriend
Preliminaries Sensitivity Analysis and Stochastic Optimization of Discrete Event Static Systems (DESS) and Sensitivity analysis and stochastic optimization ofdiscrete Event Dynamic systems (DEDS).
South African Statistical JournalGeneralized least squares estimators in the analysis of covariance structures
336 Citations1974Michael W. Browne
Mathematical ProgrammingFirst and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems
256 Citations1979H. Maurer, Jochem Zowe
Mathematical ProgrammingComplementarity and nondegeneracy in semidefinite programming
249 Citations1997Farid Alizadeh, Jean-Pierre A. Haeberly +1 more
It is shown that primal and dual nondegeneracy and strict complementarity all hold generically and Numerical experiments suggest probability distributions for the ranks ofX andZ which are consistent with the nondEGeneracy conditions.
Applied Mathematics & OptimizationMetric regularity, tangent sets, and second-order optimality conditions
249 Citations1990Roberto Cominetti
Lecture notes in control and information sciencesSecond-order necessary and sufficient optimality conditions for infinite-dimensional programming problems
218 Citations2005H. Maurer, Jochem Zowe
Mathematical ProgrammingAn envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems
138 Citations1988Hidefumi Kawasaki
Second-order necessary conditions, which involve a new term besides the second derivative of the Lagrange function, are presented, which apply to minimizing problems of sup-type functions.
SIAM Journal on Control and OptimizationSensitivity Analysis of Nonlinear Programs and Differentiability Properties of Metric Projections
135 Citations1988Alexander Shapiro
SIAM Journal on Applied MathematicsFirst Order Conditions for General Nonlinear Optimization
131 Citations1976Stephen M. Robinson
Mathematical programming studiesOn regularity conditions in mathematical programming
121 Citations1982Jean‐Paul Penot
SIAM Journal on OptimizationOn Eigenvalue Optimization
109 Citations1995Alexander Shapiro, Michael K.H. Fan
A general framework for a smooth (differentiable) approach to optimization problems involving eigenvalues of symmetric matrices is presented, based on the concept of transversality borrowed from differential geometry.
PsychometrikaRank-Reducibility of a Symmetric Matrix and Sampling Theory of Minimum Trace Factor Analysis
108 Citations1982Alexander Shapiro
Journal of Optimization Theory and ApplicationsOn the existence and nonexistence of Lagrange multipliers in Banach spaces
107 Citations1976S. Kurcyusz
SIAM Journal on Matrix Analysis and ApplicationsSecond Derivatives for Optimizing Eigenvalues of Symmetric Matrices
92 Citations1995Michael L. Overton, Robert S. Womersley
The main idea is to minimize the maximum eigen value subject to a constraint that this eigenvalue has a certain multiplicity, and the manifold $\Omega$ of matrices with such multiple eigenvalues is parameterized using a matrix exponential representation, leading to the definition of an appropriate Lagrangian function.
SIAM Journal on OptimizationOn Uniqueness of Lagrange Multipliers in Optimization Problems Subject to Cone Constraints
61 Citations1997Alexander Shapiro
This paper gives sufficient and in some cases necessary conditions for uniqueness of Lagrange multipliers in general Banach spaces and applies these conditions to two particular examples of the semidefinite and semi-infinite programming problems, respectively.
Applied Mathematics & OptimizationPerturbation theory of nonlinear programs when the set of optimal solutions is not a singleton
45 Citations1988Alexander Shapiro
Linear Algebra and its ApplicationsExtremal problems on the set of nonnegative definite matrices
42 Citations1985Alexander Shapiro
The duality result of Olkin and Pukelsheim (1982) is extended to a wide class of problems of minimizing a real-valued function of certain elements of a symmetric matrix subject to this matrix being nonnegative definite.
Mathematical ProgrammingDirectional differentiability of the optimal value function in convex semi-infinite programming
16 Citations1995Alexander Shapiro
If the unperturbed semi-infinite programming problem is convex, then the corresponding optimal value function is directionally differentiable under mild regularity assumptions, and a max-min formula for the directional derivatives is given.
Linear Algebra and its ApplicationsOn the unsolvability of inverse eigenvalues problems almost everywhere
11 Citations1983Alexander Shapiro
Second Order Necessary and Sufficient Optimality Conditions under Abstract Constraints
6 Citations1996J. Frédéric Bonnans, Roberto Cominetti +1 more
This paper introduces a condition, called second order regularity, under which there is no gap between the corresponding second order necessary and second order sufficient conditions.