We define the minimal time function associated with a collection of sets which is motivated by the optimal time problem for nonconvex constant dynamics. We first provide various basic properties of this new function: lower semicontinuity, principle of optimality, convexity, Lipschitz continuity, among others. We also compute and estimate proximal, Fréchet and limiting subdifferentials of the new function at points inside the target set as well as at points outside the target. An application to location problems is also given.
Keywords: Convex dynamics set, minimal time function, subdifferentials, normal cones, location problems
@article{COCV_2020__26_1_A93_0,
author = {Nguyen, Luong V. and Qin, Xiaolong},
title = {The minimal time function associated with a collection of sets},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2020},
publisher = {EDP Sciences},
volume = {26},
doi = {10.1051/cocv/2020017},
mrnumber = {4175377},
zbl = {1459.49008},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2020017/}
}
TY - JOUR AU - Nguyen, Luong V. AU - Qin, Xiaolong TI - The minimal time function associated with a collection of sets JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2020017/ DO - 10.1051/cocv/2020017 LA - en ID - COCV_2020__26_1_A93_0 ER -
%0 Journal Article %A Nguyen, Luong V. %A Qin, Xiaolong %T The minimal time function associated with a collection of sets %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2020017/ %R 10.1051/cocv/2020017 %G en %F COCV_2020__26_1_A93_0
Nguyen, Luong V.; Qin, Xiaolong. The minimal time function associated with a collection of sets. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 93. doi: 10.1051/cocv/2020017
[1] and , Perturbation Analysis of Optimization Problems. Springer, New York (2000). | MR | Zbl | DOI
[2] , Directional Lipschitzness of minimal time functions in Hausdorff topological vector spaces. Set-Valued Var. Anal. 22 (2014) 221–245. | MR | Zbl | DOI
[3] , On subdifferentials of a minimal time function in Hausdorff topological vector spaces. Appl. Anal. 93 (2014) 1761–1791. | MR | Zbl | DOI
[4] and , Convexity properties of the minimum time function. Calc. Var. Partial Differ. Equ. 3 (1995) 273–298. | MR | Zbl | DOI
[5] and , Semiconcave Functions Hamilton-Jacobi Equations, and Optimal Control. Birkhauser, Boston (2004). | MR | Zbl | DOI
[6] and , On the structure of the minimum time function. SIAM J. Control Optim. 48 (2010) 4776–4814. | MR | Zbl | DOI
[7] and , Differentiability properties of the minimum time function for normal linear systems. J. Math. Anal. Appl. 429 (2015) 143–174. | MR | Zbl | DOI
[8] and , Variational analysis for a class of minimal time functions in Hilbert spaces. J. Convex Anal. 11 (2004) 335–361. | MR | Zbl
[9] and , The subgradient formula for the minimal time function in the case of constant dynamics in Hilbert space. J. Global Optim. 28 (2004) 269–282. | MR | Zbl | DOI
[10] , and , Some new regularity properties for the minimal time function. SIAM J. Control Optim. 44 (2006) 2285–2299. | MR | Zbl | DOI
[11] , and , Well-posedness of minimal time problems with constant dynamics in Banach spaces. Set-Valued Var. Anal. 18 (2010) 349–372. | MR | Zbl | DOI
[12] , and , Non-Lipschitz points and the SBV regularity of the minimum time function. Calc. Var. Partial Differ. Equ. 51 (2014) 439–63. | MR | Zbl | DOI
[13] and , Vectorial penalization for generalized functional constrained problems. J. Global Optim. 68 (2017) 899–923. | MR | Zbl | DOI
[14] , and , Minimal time function with respect to a set of directions. Basic properties and applications. Optim. Methods Softw. 31 (2016) 535–61. | MR | Zbl | DOI
[15] , and , A new type of directional regularity for mappings and applications to optimization. SIAM J. Optim. 27 (2017) 1204–1229. | MR | Zbl | DOI
[16] and , Local regularity of the minimum time function. J. Optim. Theory Appl. 164 (2015) 68–91. | MR | Zbl | DOI
[17] and , Subdifferentials of a minimum time function in Banach spaces. J. Math. Anal. Appl. 321 (2006) 896–910. | MR | Zbl | DOI
[18] and , Subdifferentials of a minimal time function in normed spaces. J. Math. Anal. Appl. 358 (2009) 410–418. | MR | Zbl | DOI
[19] , Variational Analysis and Generalized Differentiation I and II, Vol. 330 and 331 in Comprehensive Studies in Mathematics. Springer, New York (2005). | Zbl
[20] and , Limiting subgradients of minimal time functions in Banach spaces. J. Global Optim. 46 (2010) 615–633. | MR | Zbl | DOI
[21] and , Subgradients of minimal time functions under minimal requirements. J. Convex Anal. 18 (2011) 915–947. | MR | Zbl
[22] and , Applications of variational analysis to a generalized Fermat - Torricelli problem. J. Optim. Theory Appl. 148 (2011) 431–454. | MR | Zbl | DOI
[23] , and , Applications of variational analysis to a generalized Heron problem. Appl. Anal. 91 (2012) 1915–1942. | MR | Zbl | DOI
[24] and , A generalized Sylvester problem and a generalized Fermat-Torricelli problem. J. Convex Anal. 20 (2013) 669–687. | MR | Zbl
[25] and , Variational analysis of directional minimal time functions and applications to location problems. Set-Valued Var. Anal. 21 (2013) 405–430. | MR | Zbl | DOI
[26] and , Subgradients of minimal time functions without calmness. J. Convex Anal. 26 (2019) 189–200. | MR | Zbl
[27] , and , Minimal time functions and the smallest intersecting ball problem with unbounded dynamics. J. Optim. Theory Appl. 154 (2012) 768–791. | MR | Zbl | DOI
[28] , , and , Nonsmooth algorithms and Nesterov’s smoothing techniques for generalized Fermat-Torricelli problems. SIAM J. Optim. 24 (2014) 1815–1839. | MR | Zbl | DOI
[29] , Variational analysis and regularity of the minimum time function for differential inclusions. SIAM J. Control Optim. 54 (2016) 2235–2258. | MR | Zbl | DOI
[30] and , On variational analysis for general distance functions. In prepraration.
[31] and , Exact characterztion for subdifferentials a speccial optimal value function. Optim. Lett. 12 (2018) 519–534. | MR | Zbl | DOI
[32] and , Proximal analysis and the minimal time function. SIAM J. Control Optim. 36 (1998) 1048–1072. | MR | Zbl | DOI
Cité par Sources :
This paper was supported by the National Natural Science Foundation of China under Grant No.11401152.
LVN was supported by the Research Fund for International Young Scientists from NNSFC under Grant No.11850410438 and China Postdoctoral Science Foundation under Grant No. 2017M6200421.
