Stable well-posedness and tilt stability with respect to admissible functions

Zheng, Xi Yin; Zhu, Jiangxing

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1397-1418.

Note that the well-posedness of a proper lower semicontinuous function $f$ can be equivalently described using an admissible function. In the case when the objective function $f$ undergoes the tilt perturbations in the sense of Poliquin and Rockafellar, adopting admissible functions $\varphi$ and $\psi$, this paper introduces and studies the stable well-posedness of $f$ with respect to $\varphi$ (in brief, $\varphi$-SLWP) and tilt-stable local minimum of $f$ with respect to $\psi$ (in brief, $\psi$-TSLM). In the special case when $\varphi \left(t\right)=t^2$ and $\psi \left(t\right)=t$, the corresponding $\varphi$-SLWP and $\psi$-TSLM reduce to the stable second order local minimizer and tilt stable local minimum respectively, which have been extensively studied in recent years. We discover an interesting relationship between two admissible functions $\varphi$ and $\psi$: $\psi \left(t\right)={(\varphi \text{'})}^{-1}\left(t\right)$, which implies that a proper lower semicontinuous function $f$ on a Banach space has $\varphi$-SLWP if and only if $f$ has $\psi$-TSLM. Using the techniques of variational analysis and conjugate analysis, we also prove that the strong metric $\varphi \text{'}$-regularity of $\partial f$ is a sufficient condition for $f$ to have $\varphi$-SLWP and that the strong metric $\varphi \text{'}$-regularity of $\partial [\overline{co}$$(f+\delta$${}_{B_X[\overline{x},r]}\left)\right]$ for some $r>0$ is a necessary condition for $f$ to have $\varphi$-SLWP. In the special case when $\varphi \left(t\right)=t^2$, our results cover some existing main results on the tilt stability.

F.J. Aragón Artacho and M.H. Geoffroy, Characterization of metric regularity of subdifferentials. J. Convex Anal. 15 (2008) 365–380. | MR | Zbl

H. Attouch and R.J.-B. Wets, Quantitative stability of variational systems ii, a framewrok for nonlinear conditioning. SIAM J. Optim. 3 (1993) 359–381. | DOI | MR | Zbl

E. Bednarczuk, Weak sharp efficiency and growth condition for vector-valued functions with applications. Optimization 53 (2004) 455–474. | DOI | MR | Zbl

F. J. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems. New York (2000). | MR | Zbl

J.V. Burke and M.C. Ferris, Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31 (1993) 1340–1359. | DOI | MR | Zbl

F.H. Clarke, Optimization and Nonsmooth Analysis. Wiley, New York (1983). | MR | Zbl

A.L. Dontchev and R. Rockafellar, Regularity and conditioning of solution mappings in variational analysis. Set-Valued Anal. 12 (2004) 79–109. | DOI | MR | Zbl

A.L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems, in Vol. 1543 of Lecture Notes in Mathematics. Springer-Verlag (1993). | MR | Zbl

D. Drusvyatskiy and A.S. Lewis, Tilt stability, uniform quadratic growth, and strong metric regularity of the subdifferential. SIAM J. Optim. 23 (2013) 256–267. | DOI | MR | Zbl

D. Drusvyatskiy, B.S. Mordukhovich and T.T.A. Nghia, Second-order growth, tilt stability, and metric regularity of the subdifferential. J. Convex Anal. 21 (2014) 1165–1192. | MR | Zbl

M. Ferris, Weak sharp minima and penalty fucntions in mathematical programming. Ph.D. thesis, University of Cambridge (1988).

H. Frankowska and M. Quincampoix, Hölder metric regularity of set-valued maps. Math. Program. 132 (2012) 333–354. | DOI | MR | Zbl

X.X. Huang and X.Q. Yang, Generalized levitin-polyak well-posedness in constrained optimization. SIAM J. Optim. 17 (2006) 243–258. | DOI | MR | Zbl

A.D. Ioffe, Metric regularity and subdifferential calculus. Russ. Math. Surv. 55 (2000) 501–558. | DOI | MR | Zbl

A. Jourani, L. Thibault and D. Zagrodny, $c^{1,\omega (·)}$-regularity and lipschitz-like properties of subdifferential. Proc. London Math. Soc. 105 (2012) 189–223. | DOI | MR | Zbl

P. Kenderov, Semi-continuity of set-valued monotone mappings. Fund. Math. 88 (1975) 61–69. | DOI | MR | Zbl

R. Lucchetti, Convexity and Well-posedness Problems. CMS Books in Mathematics. Springer, New York (2006). | MR | Zbl

B.S. Mordukhovich, Variational Analysis and Generalized differentiation I. Berlin Heidelberg (2006). | Zbl

B.S. Modukhovich and J.V. Outrata, Tilt stability in nonlinear programming under mangasarian-fromovitz constraint qualification. Kybernetika 49 (2013) 446–464. | MR | Zbl

B.S. Mordukhovich and T.A. Nghia, Second-order variational analysis and characterizations of tilt-stable optimal solutions in infinite-dimensional spaces. Nonlinear Anal. 86 (2013) 159–180. | DOI | MR | Zbl

B.S. Mordukhovich and T.T.A. Nghia, Full lipschitzian and hölder stability in optimization with applications to mathematical programming and optimal control. SIAM J. Optim. 24 (2014) 1344–1381. | DOI | MR | Zbl

B.S. Mordukhovich and T.T.A. Nghia, Second-order characterizations of tilt stability with applications to nonlinear optimization. Math. Program. 149 (2015) 83–104. | DOI | MR | Zbl

R.R. Phelps, Convex Functions, Monotone Operators and Differentiability. Vol. 1364 of Lecture Notes in Math. Springer, New York (1989). | MR | Zbl

R.A. Poliquin and R.T. Rockafellar, Tilt stability of a local minimum. SIAM J. Optim. 8 (1998) 287–299. | DOI | MR | Zbl

J.P. Revalski, Hadamard and strong well-posedness for convex programs. SIAM J. Optim. 7 (1997) 519–526. | DOI | MR | Zbl

W. Schirotzek, Nonsmooth Analysis, Berlin-Heidelberg-New York (2007). | MR | Zbl

M. Studniarski and D.E. Ward, Weak sharp minima: Characterizations and sufficient conditions. SIAM J. Control Optim. 38 (1999) 219–236. | DOI | MR | Zbl

J.C. Yao and X.Y. Zheng, Error bound and well-posedness with respect to an admissible function. Applicable Anal. 95 (2016) 1070–1087. | DOI | MR | Zbl

X.Y. Zheng and K.F. Ng, Metric regularity and constraint qualifications for convex inequalities on banach sapces. SIAM J. Optim. 14 (2003) 757–772. | DOI | MR | Zbl

X.Y. Zheng and K.F. Ng, Metric subregularity and calmness for nonconvex generalized equations in banach spaces. SIAM J. Optim. 20 (2010) 2119–2136. | DOI | MR | Zbl

X.Y. Zheng and K.F. Ng, Hölder weak sharp minimizers and hölder tilt-stability. Nonlin. Anal. 120 (2015) 186–201. | DOI | MR | Zbl

X.Y. Zheng and K.F. Ng, Hölder stable minimizers, tilt stability and hölder metric regularity of subdifferential. SIAM J. Optim. 25 (2015) 416–438. | DOI | MR | Zbl

X.Y. Zheng and X.Q. Yang, Weak sharp minima for semi-infinite optimization problems with applications. SIAM J. Optim. 18 (2007) 573–588. | DOI | MR | Zbl

C. Zălinescu, Convex analysis in general vector spaces. World Scientific (2002). | MR | Zbl