Generalized Sensitivity Analysis of Nonlinear Programs

This paper extends classical sensitivity results for nonlinear programs to cases in which parametric perturbations cause changes in the active set. This is accomplished using lexicographic directional derivatives, a recently developed tool in nonsmooth analysis based on Nesterov's lexicographic differentiation. A nonsmooth implicit function theorem is augmented with generalized derivative information and applied to a standard nonsmooth reformulation of the parametric KKT system. It is shown that the sufficient conditions for this implicit function theorem variant are implied by a KKT point satisfying the linear independence constraint qualification and strong second-order sufficiency. Mirroring the classical theory, the resulting sensitivity system is a nonsmooth equation system which admits primal and dual sensitivities as its unique solution. Practically implementable algorithms are provided for calculating the nonsmooth sensitivity system's unique solution, which is then used to furnish B-subdifferenti...

[1]  Stephan Dempe,et al.  Directional differentiability of optimal solutions under Slater's condition , 1993, Math. Program..

[2]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[3]  A. Shapiro.  Sensitivity analysis of nonlinear programs and differentiability properties of metric projections , 1988 .

[4]  Yurii Nesterov,et al.  Lexicographic differentiation of nonsmooth functions , 2005, Math. Program..

[5]  Paul I. Barton,et al.  Generalized Derivatives for Hybrid Systems , 2017, IEEE Transactions on Automatic Control.

[6]  Defeng Sun,et al.  On NCP-Functions , 1999, Comput. Optim. Appl..

[7]  Jerzy Kyparisis,et al.  Sensitivity Analysis for Nonlinear Programs and Variational Inequalities with Nonunique Multipliers , 1990, Math. Oper. Res..

[8]  B. Mordukhovich.  Generalized Differential Calculus for Nonsmooth and Set-Valued Mappings , 1994 .

[9]  Paul I. Barton,et al.  Generalized derivatives of optimal control problems with nonsmooth differential-algebraic equations embedded , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[10]  Francisco Facchinei,et al.  An LP-Newton method: nonsmooth equations, KKT systems, and nonisolated solutions , 2013, Mathematical Programming.

[11]  Andreas Griewank,et al.  Evaluating derivatives - principles and techniques of algorithmic differentiation, Second Edition , 2000, Frontiers in applied mathematics.

[12]  Paul I. Barton,et al.  A vector forward mode of automatic differentiation for generalized derivative evaluation , 2015, Optim. Methods Softw..

[13]  Stephen M. Robinson,et al.  Strongly Regular Generalized Equations , 1980, Math. Oper. Res..

[14]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[15]  Paul I. Barton,et al.  Reliable Flash Calculations: Part 1. Nonsmooth Inside-Out Algorithms , 2017 .

[16]  R. Janin.  Directional derivative of the marginal function in nonlinear programming , 1984 .

[17]  Paul I. Barton,et al.  Generalized gradient elements for nonsmooth optimal control problems , 2014, 53rd IEEE Conference on Decision and Control.

[18]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[19]  J. Frédéric Bonnans,et al.  Perturbation Analysis of Optimization Problems , 2000, Springer Series in Operations Research.

[20]  B. Mordukhovich,et al.  Coderivatives in parametric optimization , 2004, Math. Program..

[21]  James Hardy Wilkinson,et al.  Rounding errors in algebraic processes , 1964, IFIP Congress.

[22]  Diethard Klatte,et al.  Nonsmooth Equations in Optimization: "Regularity, Calculus, Methods And Applications" , 2006 .

[23]  Daniel Ralph,et al.  Sensitivity analysis of composite piecewise smooth equations , 1997, Math. Program..

[24]  Wolfgang Marquardt,et al.  Fast NMPC schemes for regulatory and economic NMPC – A review ☆ , 2016 .

[25]  T. H. III Sweetser,et al.  A minimal set-valued strong derivative for vector-valued Lipschitz functions , 1977 .

[26]  C. Lemaréchal,et al.  ON A BUNDLE ALGORITHM FOR NONSMOOTH OPTIMIZATION , 1981 .

[27]  Liqun Qi,et al.  A nonsmooth version of Newton's method , 1993, Math. Program..

[28]  J. Cortés.  Discontinuous dynamical systems , 2008, IEEE Control Systems.

[29]  M. Kojima.  Strongly Stable Stationary Solutions in Nonlinear Programs. , 1980 .

[30]  B. Mordukhovich.  Variational analysis and generalized differentiation , 2006 .

[31]  Paul I. Barton,et al.  Generalized Derivatives of Lexicographic Linear Programs , 2018, J. Optim. Theory Appl..

[32]  Stephan Dempe,et al.  Directional derivatives of the solution of a parametric nonlinear program , 1995, Math. Program..

[33]  Yu. S. Ledyaev,et al.  Nonsmooth analysis and control theory , 1998 .

[34]  Paul I. Barton,et al.  Generalized Derivatives for Solutions of Parametric Ordinary Differential Equations with Non-differentiable Right-Hand Sides , 2014, J. Optim. Theory Appl..

[35]  Anthony V. Fiacco,et al.  Introduction to Sensitivity and Stability Analysis in Nonlinear Programming , 2012 .

[36]  P. I. Barton,et al.  Reliable Flash Calculations: Part 2. Process flowsheeting with nonsmooth models and generalized derivatives , 2017 .

[37]  Paul I. Barton,et al.  Dependence of solutions of nonsmooth differential-algebraic equations on parameters , 2017 .

[38]  Jong-Shi Pang,et al.  Piecewise Smoothness, Local Invertibility, and Parametric Analysis of Normal Maps , 1996, Math. Oper. Res..

[39]  Kamil A. Khan.  Branch-locking AD techniques for nonsmooth composite functions and nonsmooth implicit functions , 2018, Optim. Methods Softw..

[40]  S. Scholtes.  Introduction to Piecewise Differentiable Equations , 2012 .

[41]  Paul I. Barton,et al.  Generalized derivatives of dynamic systems with a linear program embedded , 2016, Autom..

[42]  Paul I. Barton,et al.  Computationally relevant generalized derivatives: theory, evaluation and applications , 2018, Optim. Methods Softw..