The adjoint arc in nonsmooth optimization

Loewen, Philip D.; Rockafellar, R. T.

doi:10.1090/S0002-9947-1991-1036004-7

The adjoint arc in nonsmooth optimization
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by Philip D. Loewen and R. T. Rockafellar PDF

Trans. Amer. Math. Soc. 325 (1991), 39-72 Request permission

Abstract:

We extend the theory of necessary conditions for nonsmooth problems of Bolza in three ways: first, we incorporate state constraints of the intrinsic type $x(t) \in X(t)$ for all $t$; second, we make no assumption of calmness or normality; and third, we show that a single adjoint function of bounded variation simultaneously satisfies the Hamiltonian inclusion, the Euler-Lagrange inclusion, and the Weierstrass-Pontryagin maximum condition, along with the usual transversality relations.

References

Jean-Pierre Aubin and Ivar Ekeland, Applied nonlinear analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 749753
Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
F. H. Clarke, Optimal solutions to differential inclusions, J. Optim. Theory Appl. 19 (1976), no. 3, 469–478. MR 425714, DOI 10.1007/BF00941488
Frank H. Clarke, Extremal arcs and extended Hamiltonian systems, Trans. Amer. Math. Soc. 231 (1977), no. 2, 349–367. MR 442784, DOI 10.1090/S0002-9947-1977-0442784-4
Frank H. Clarke, The generalized problem of Bolza, SIAM J. Control Optim. 14 (1976), no. 4, 682–699. MR 412926, DOI 10.1137/0314044
F. H. Clarke, Hamiltonian analysis of the generalized problem of Bolza, Trans. Amer. Math. Soc. 301 (1987), no. 1, 385–400. MR 879580, DOI 10.1090/S0002-9947-1987-0879580-6
Philip D. Loewen, Frank H. Clarke, and Richard B. Vinter, Differential inclusions with free time, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 6, 573–593 (English, with French summary). MR 978672
Philip D. Loewen, The proximal normal formula in Hilbert space, Nonlinear Anal. 11 (1987), no. 9, 979–995. MR 907818, DOI 10.1016/0362-546X(87)90079-4

Analyse proximale en optimisation

R. Tyrrell Rockafellar, Existence theorems for general control problems of Bolza and Lagrange, Advances in Math. 15 (1975), 312–333. MR 365273, DOI 10.1016/0001-8708(75)90140-1

Convex analysis

R. Tyrrell Rockafellar, State constraints in convex control problems of Bolza, SIAM J. Control 10 (1972), 691–715. MR 0324505
R. Tyrrell Rockafellar, Dual problems of Lagrange for arcs of bounded variation, Calculus of variations and control theory (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975; dedicated to Laurence Chisholm Young on the occasion of his 70th birthday), Publ. Math. Res. Center Univ. Wisconsin, No. 36, Academic Press, New York, 1976, pp. 155–192. MR 0482468

Optimality conditions for convex control problems with nonnegative states and the possibility of jumps

R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, Math. Programming Stud. 17 (1982), 28–66. MR 654690, DOI 10.1007/bfb0120958
R. Tyrrell Rockafellar, Integral functionals, normal integrands and measurable selections, Nonlinear operators and the calculus of variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975) Lecture Notes in Math., Vol. 543, Springer, Berlin, 1976, pp. 157–207. MR 0512209
R. T. Rockafellar, Integrals which are convex functionals. II, Pacific J. Math. 39 (1971), 439–469. MR 310612
R. T. Rockafellar, Extensions of subgradient calculus with applications to optimization, Nonlinear Anal. 9 (1985), no. 7, 665–698. MR 796082, DOI 10.1016/0362-546X(85)90012-4
R. T. Rockafellar, Hamiltonian trajectories and duality in the optimal control of linear systems with convex costs, SIAM J. Control Optim. 27 (1989), no. 5, 1007–1025. MR 1009335, DOI 10.1137/0327054