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78
LAGRANGE MULTIPLIERS AND OPTIMALITY
, 1993
"... Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions ..."
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Cited by 114 (7 self)
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Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger theoretical picture. A major line of research has been the nonsmooth geometry of onesided tangent and normal vectors to the set of points satisfying the given constraints. Another has been the gametheoretic role of multiplier vectors as solutions to a dual problem. Interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored. Lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows blackandwhite constraints to be replaced by penalty expressions. This paper traces such themes in the current theory of Lagrange multipliers, providing along the way a freestanding exposition of basic nonsmooth analysis as motivated by and applied to this subject.
Characterizations of strong regularity for variational inequalities over polyhedral convex sets
 SIAM J. OPTIMIZATION
, 1996
"... Linear and nonlinear variational inequality problems over a polyhedral convex set are analyzed parametrically. Robinson’s notion of strong regularity, as a criterion for the solution set to be a singleton depending Lipschitz continuously on the parameters, is characterized in terms of a new “critica ..."
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Cited by 60 (16 self)
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Linear and nonlinear variational inequality problems over a polyhedral convex set are analyzed parametrically. Robinson’s notion of strong regularity, as a criterion for the solution set to be a singleton depending Lipschitz continuously on the parameters, is characterized in terms of a new “critical face” condition and in other ways. The consequences for complementarity problems are worked out as a special case. Application is also made to standard nonlinear programming problems with parameters that include the canonical perturbations. In that framework a new characterization of strong regularity is obtained for the variational inequality associated with the KarushKuhnTucker conditions.
Numerical solution of optimal control problems by direct collocation’, Ed
 In Optimal Control  Calculus of Variation, Optimal Control Theory and Numerical Methods, 111, 129143, International Series of Numerical Mathematics, Birkhäuser
, 1993
"... By an appropriate discretization of control and state variables, a constrained optimal control problem is transformed into a finite dimensional nonlinear program which can be solved by standard SQPmethods [10]. Convergence properties of the discretization are derived. From a solution of this method ..."
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Cited by 56 (2 self)
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By an appropriate discretization of control and state variables, a constrained optimal control problem is transformed into a finite dimensional nonlinear program which can be solved by standard SQPmethods [10]. Convergence properties of the discretization are derived. From a solution of this method known as direct collocation, these properties are used to obtain reliable estimates of adjoint variables. In the presence of active state constraints, these estimates can be significantly improved by including the switching structure of the state constraint into the optimization procedure. Two numerical examples are presented. 1 Statement of problems Systems governed by ordinary differential equations arise in many applications as, e. g., in astronautics, aeronautics, robotics, and economics. The task of optimizing these systems leads to the optimal control problems investigated in this paper. The aim is to find a control vector u(t) and the final time tf that minimize the functional
Nonsmooth Lagrangian mechanics and variational collision integrators
 SIAM Journal on Applied Dynamical Systems
, 2003
"... Abstract. Variational techniques are used to analyze the problem of rigidbody dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions, and it is shown in what sense the system is symplectic and satisfies a Noetherstyle momentum ..."
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Cited by 35 (8 self)
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Abstract. Variational techniques are used to analyze the problem of rigidbody dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for collisions, and it is shown in what sense the system is symplectic and satisfies a Noetherstyle momentum conservation theorem. Discretizations of this nonsmooth mechanics are developed by using the methodology of variational discrete mechanics. This leads to variational integrators which are symplecticmomentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the longtime stable energy behavior typical of variational methods is demonstrated.
order sufficient conditions for time–optimal bang–bang control problems
 SIAM J. Control and Optimization
"... Abstract. We study second order sufficient optimality conditions (SSC) for optimal control problems with control appearing linearly. Specifically, timeoptimal bangbang controls will be in ..."
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Cited by 20 (5 self)
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Abstract. We study second order sufficient optimality conditions (SSC) for optimal control problems with control appearing linearly. Specifically, timeoptimal bangbang controls will be in
Maximum entropy reconstruction using derivative information part 2: . . .
, 1995
"... Maximum entropy density estimation, a technique for reconstructing an unknown density function on the basis of certain measurements, has applications in various areas of applied physical sciences and engineering. Here we present numerical results for the maximum entropy inversion program based on a ..."
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Cited by 15 (3 self)
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Maximum entropy density estimation, a technique for reconstructing an unknown density function on the basis of certain measurements, has applications in various areas of applied physical sciences and engineering. Here we present numerical results for the maximum entropy inversion program based on a new class of information measures which are designed to control derivative values of the unknown densities.
Second Order Sufficient Conditions for Optimal Control Problems with Free Final Time: The Riccati Approach
, 2000
"... Second order sufficient conditions (SSC) for control problems with controlstate constraints and free final time are presented. Instead of deriving such SSC de initio, the control problem with free final time is tranformed into an augmented control problem with fixed final time for which wellknown ..."
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Cited by 14 (4 self)
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Second order sufficient conditions (SSC) for control problems with controlstate constraints and free final time are presented. Instead of deriving such SSC de initio, the control problem with free final time is tranformed into an augmented control problem with fixed final time for which wellknown SSC exist. SSC are then expressed as a condition on the positive definiteness of the second variation. A convenient numerical tool for verifying this condition is based on the Riccati approach where one has to find a bounded solution of an associated Riccati equation satisfying specific boundary conditions. The augmented Riccati equations for the augmented control problem are derived and their modifications on the boundary of the controlstate constraint are discussed. Two numerical examples, (1) the classical EarthMars orbit transfer in minimal time, (2) the Rayleigh problem in electrical engineering, demonstrate that the Riccati equation approach provides a viable numerical test of SSC.
New necessary conditions for the generalized problem of Bolza
 SIAM J. Control Optim
, 1996
"... Abstract. Problems of optimal control are considered in the neoclassical Bolza format, which centers on states and velocities and relies on nonsmooth analysis. Subgradient versions of the EulerLagrange equation and the Hamiltonian equation are shown to be necessary for the optimality of a trajector ..."
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Cited by 12 (3 self)
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Abstract. Problems of optimal control are considered in the neoclassical Bolza format, which centers on states and velocities and relies on nonsmooth analysis. Subgradient versions of the EulerLagrange equation and the Hamiltonian equation are shown to be necessary for the optimality of a trajectory, moreover in a newly sharpened form that makes these conditions equivalent to each other. At the same time, the assumptions on the Lagrangian integrand are weakened substantially over what has been required previously in obtaining such conditions. Key words. Optimal control, calculus of variations, nonsmooth analysis, problem of Bolza, EulerLagrange condition, Hamiltonian condition, transversality condition AMS subject classifications. 49K15, 49K05, 49K24 1. Introduction. Among
Regularity and variationality of solutions to HamiltonJacobi equations. part i: regularity
 ESAIM Control Optim. Calc. Var
"... We formulate an Hamilton–Jacobi partial differential equation H(x, Du(x)) = 0 on a n dimensional manifold M, with assumptions of convexity of the sets {p: H(x, p) ≤ 0} ⊂ T ∗ x M, for all x. In this paper we reduce the above problem to a simpler problem: this shows that u may be built using an asy ..."
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Cited by 11 (4 self)
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We formulate an Hamilton–Jacobi partial differential equation H(x, Du(x)) = 0 on a n dimensional manifold M, with assumptions of convexity of the sets {p: H(x, p) ≤ 0} ⊂ T ∗ x M, for all x. In this paper we reduce the above problem to a simpler problem: this shows that u may be built using an asymmetric distance (this is a generalization of the “distance function ” in Finsler Geometry): this brings forth a ’completeness ’ condition, and a Hopf–Rinow theorem adapted to Hamilton–Jacobi problems. The ’completeness’ condition implies that u is the unique viscosity solution to the above problem. When H is moreover of class C 1,1, we show how the completeness condition is equivalent to a condition expressed using the characteristics equations. 7