×

The Hopf-Lax formula for multiobjective costs with non-constant discount via set optimization. (English) Zbl 1505.49016

In this paper, the minimization of a multiobjective Lagrangian with non-constant discount is studied. To this end, the problem is embedded into a set-valued framework, and a Bellman’s optimality principle and a Hopf-Lax formula are derived. The value function is shown to be a solution of a set-valued Hamilton-Jacobi equation.

MSC:

49J53 Set-valued and variational analysis
35F21 Hamilton-Jacobi equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aubin, J.-P., Lax-Hopf formula and Max-Plus properties of solutions to Hamilton-Jacobi equations, NoDEA Nonlinear Differ. Equ. Appl., 20, 187-211 (2013) · Zbl 1269.49023
[2] Aubin, J.-P.; Frankowska, H., Set-Valued Analysis (1990), Birkhäuser: Birkhäuser Boston-Basel-Berlin · Zbl 0713.49021
[3] Aumann, R. J., Integrals of set-valued functions, J. Math. Anal. Appl., 12, 1-12 (1965) · Zbl 0163.06301
[4] Avantaggiati, A.; Loreti, P., Lax type formulas with lower semicontinuous initial data and hypercontractivity results, NoDEA Nonlinear Differ. Equ. Appl., 20, 385-411 (2013) · Zbl 1275.35079
[5] Barro, R., Ramsey meets laibson in the neoclassical growth model, Q. J. Econ., 114, 1125-1152 (1999) · Zbl 0940.91056
[6] Barron, E. N.; Jensen, R.; Liu, W., Hopf-Lax-type formula for \(u_t + H(u, D u) = 0\), J. Differ. Equ., 126, 48-61 (1996) · Zbl 0857.35023
[7] Boyd, S. P.; Vandenberghe, L., Convex Optimization, 50-51 (2004), Cambridge University Press · Zbl 1058.90049
[8] De Marco, G.; Gorni, G.; Zampieri, G., Global inversion of functions: an introduction, Nonlinear Differ. Equ. Appl., 1, 229-248 (1994) · Zbl 0820.58008
[9] Evans, G. C., Mathematical Introduction to Economics (1930), McGraw-Hill · JFM 56.1108.05
[10] Hamel, A. H.; Heyde, F.; Löhne, A.; Rudloff, B.; Schrage, C., Set optimization – a rather short introduction, (Hamel, A. H.; Heyde, F.; Löhne, A.; Rudloff, B.; Schrage, C., Set Optimization and Applications - The State of the Art (2015), Springer), 65-141 · Zbl 1337.49001
[11] Hamel, A. H.; Schrage, C., Directional derivatives, subdifferentials and optimality conditions for set-valued convex functions, Pac. J. Optim., 10, 4, 667-689 (2014) · Zbl 1307.90199
[12] Hamel, A. H.; Visetti, D., The value functions approach and Hopf-Lax formula for multiobjective costs via set optimization, J. Math. Anal. Appl., 483, 1, Article 123605 pp. (2020) · Zbl 1429.49019
[13] Heyde, F.; Löhne, A., Solution concepts in vector optimization: a fresh look at an old story, Optimization, 60, 12, 1421-1440 (2011) · Zbl 1258.90064
[14] Hoang, N., Hopf-Lax formula and generalized characteristics, Appl. Anal., 96, 2 (2013)
[15] Hopf, E., Generalized solutions of non-linear equations of first order, J. Math. Mech., 14, 951-973 (1965) · Zbl 0168.35101
[16] Karp, L., Non-constant discounting in continuous time, J. Econ. Theory, 132, 557-568 (2007) · Zbl 1142.91668
[17] Laibson, D., Golden eggs and hyperbolic discounting, Q. J. Econ., 112, 2, 443-477 (1997) · Zbl 0882.90024
[18] Lax, P. D., Hyperbolic systems of conservation laws II, Commun. Pure Appl. Math., 10, 537-566 (1957) · Zbl 0081.08803
[19] Marín-Solano, J.; Navas, J., Non-constant discounting in finite horizon: the free terminal time case, J. Econ. Dyn. Control, 33, 666-675 (2009) · Zbl 1168.49025
[20] Marín-Solano, J.; Patxot, C., Heterogeneous discounting in economic problems, Optim. Control Appl. Methods, 33, 1, 32-50 (2012) · Zbl 1262.91086
[21] Marín-Solano, J.; Shevkoplyas, E. V., Non-constant discounting and differential games with random time horizon, Automatica, 47, 2626-2638 (2011) · Zbl 1235.49074
[22] Pilecka, M., Set-valued optimization and its application to bilevel optimization (2016), Technische Universität Bergakademie Freiberg, PhD-thesis
[23] Rincón-Zapatero, J. P., Hopf-Lax formula for variational problems with non-constant discount, J. Geom. Mech., 1, 3, 357-367 (2009) · Zbl 1191.35101
[24] Strömberg, T., The Hopf-Lax formula gives the unique viscosity solution, Differ. Integral Equ., 15, 1, 47-52 (2002) · Zbl 1026.49023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.