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Set-Valued and Variational Analysis

, Volume 24, Issue 4, pp 685–705 | Cite as

Legendre Transform and Applications to Finite and Infinite Optimization

  • Cristopher Hermosilla
Article

Abstract

We investigate convex constrained nonlinear optimization problems and optimal control with convex state constraints in the light of the so-called Legendre transform. We use this change of coordinate to propose a gradient-like algorithm for mathematical programs, which can be seen as a search method along geodesics. We also use the Legendre transform to study the value function of a state constrained Mayer problem and we show that it can be characterized as the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.

Keywords

Convex functions of Legendre type Legendre transform Geodesic search methods Optimal control Convex state constraints Riemannian metrics 

Mathematics Subject Classification (2010)

MSC 65K05 MSC 90C51 MSC 93C15 MSC 49L25 

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References

  1. 1.
    Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2009)MATHGoogle Scholar
  2. 2.
    Álvarez, F., Bolte, J., Brahic, O.: Hessian Riemannian gradient flows in convex programming. SIAM J. Control Optim. 43(2), 477–501 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Attouch, H., Bolte, J., Redont, P., Teboulle, M.: Singular Riemannian barrier methods and gradient-projection dynamical systems for constrained optimization. Optimization 53(5–6), 435–454 (2004)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Auslender, A., Teboulle, M.: Interior gradient and proximal methods for convex and conic optimization. SIAM J. Optim. 16(3), 697–725 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. In: Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston. With appendices by Maurizio Falcone and Pierpaolo Soravia (1997)Google Scholar
  6. 6.
    Bayer, D.A., Lagarias, J.C.: The nonlinear geometry of linear programming. I. Affine and projective scaling trajectories. Trans. Am. Math. Soc. 314(2), 499–526 (1989)MathSciNetMATHGoogle Scholar
  7. 7.
    Bokanowski, O., Forcadel, N., Zidani, H.: Deterministic state-constrained optimal control problems without controllability assumptions. ESAIM: Control Optim. Calc. Var. 17(04), 995–1015 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bolte, J., Teboulle, M.: Barrier operators and associated gradient-like dynamical systems for constrained minimization problems. SIAM J. Control Optim. 42(4), 1266–1292 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Borwein, J.M., Vanderwerff, J.: Convex functions of Legendre type in general Banach spaces. J. Convex Anal. 8(2), 569–582 (2001)MathSciNetMATHGoogle Scholar
  10. 10.
    Borwein, J.M., Vanderwerff, J.D.: Convex Functions: Constructions, Characterizations and Counterexamples. Cambridge University Press, Cambridge (2010)CrossRefMATHGoogle Scholar
  11. 11.
    Boscain, U., Piccoli, B.: Optimal Syntheses for Control Systems on 2-D Manifolds. Springer, Berlin (2004)MATHGoogle Scholar
  12. 12.
    Burachik, R., Drummond, L.M.G., Iusem, A.N., Svaiter, B.F.: Full convergence of the steepest descent method with inexact line searches. Optimization 32(2), 137–146 (1995)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chryssochoos, I., Vinter, R.: Optimal control problems on manifolds: a dynamic programming approach. J. Math. Anal. Appl. 287(1), 118–140 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Clarke, F., Ledyaev, Y., Stern, R., Wolenski, P.: Nonsmooth Analysis and Control Theory, 1 edn. Springer-Verlag, New York (1998)Google Scholar
  15. 15.
    Clarke, F., Stern, R.: Hamilton-Jacobi characterization of the state constrained value. Nonlinear Anal. Theory Methods Appl. 61(5), 725–734 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Crouzeix, J.: A relationship between the second derivatives of a convex function and of its conjugate. Math. Program. 13(1), 364–365 (1977)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Boston (1992)CrossRefMATHGoogle Scholar
  18. 18.
    Fiacco, A.V.: Perturbed variations of penalty function methods. Example: projective SUMT. Ann. Oper. Res. 27(1–4), 371–380 (1990)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Frankowska, H., Mazzola, M.: Discontinuous solutions of Hamilton–Jacobi–Bellman equation under state constraints. Calc. Var. Partial Differ. Equ. 46(3-4), 725–747 (2013)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Frankowska, H., Vinter, R.B.: Existence of neighboring feasible trajectories: applications to dynamic programming for state-constrained optimal control problems. J. Optim. Theory Appl. 104(1), 20–40 (2000)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Hermosilla, C., Zidani, H.: Infinite horizon problems on stratifiable state-constraints sets. J. Differ. Equ. 258(4), 1430–1460 (2015)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Iusem, A.N., Svaiter, B.F., Da Cruz Neto, J.X.: Central paths, generalized proximal point methods, and Cauchy trajectories in Riemannian manifolds. SIAM J. Control Optim. 37(2), 566–588 (electronic) (1999)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Klötzler, R.: On a general conception of duality in optimal control. In: Equadiff IV, pp 189–196. Springer, Berlin (1979)Google Scholar
  24. 24.
    Lee, J.M.: Riemannian Manifolds: An Introduction to Curvature. Springer-Verlag, New York (1997)CrossRefMATHGoogle Scholar
  25. 25.
    Lee, J.M.: Introduction to Smooth Manifolds. Springer-Verlag, New York (2012)Google Scholar
  26. 26.
    McCormick, G.P.: The projective SUMT method for convex programming. Math. Oper. Res. 14(2), 203–223 (1989)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Nesterov, Y., Todd, M.: On the Riemannian geometry defined by self-concordant barriers and interior-point methods. Found. Comput. Math. 2(4), 333–361 (2002)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer-Verlag, New York (2006)Google Scholar
  29. 29.
    Quiroz, E.P., Oliveira, P.R.: Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds. J. Convex Anal. 16(1), 49–69 (2009)MathSciNetMATHGoogle Scholar
  30. 30.
    Renegar, J.: A Mathematical View of Interior-Point Methods in Convex Optimization. Society for Industrial and Applied Mathematics (2001)Google Scholar
  31. 31.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970)Google Scholar
  32. 32.
    Rockafellar, R.T.: Conjugate duality and optimization. Society for Industrial and Applied Mathematics (1974)Google Scholar
  33. 33.
    Smith, S.T.: Optimization techniques on Riemannian manifolds. Fields Inst. Commun. 3(3), 113–135 (1994)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Soner, H.: Optimal control with state-space constraint I. SIAM J. Control Optim. 24(3), 552–561 (1986)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Udriste, C.: Convex Function and Optimization Methods on Riemannian Manifolds. Springer, Netherlands (1994)Google Scholar
  36. 36.
    Vanderbei, R.J.: Linear programming. In: Foundations and extensions, International Series in Operations Research & Management Science, vol 37 (2001)Google Scholar
  37. 37.
    Vinter, R.: Optimal Control. Springer, Berlin (2010)CrossRefMATHGoogle Scholar
  38. 38.
    Vinter, R.: Convex duality and nonlinear optimal control. SIAM J. Control Optim. 31(2), 518–538 (1993)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Yang, Y.: Globally convergent optimization algorithms on Riemannian manifolds: uniform framework for unconstrained and constrained optimization. J. Optim. Theory Appl. 132(2), 245–265 (2007)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA

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