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33
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 98 (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.
A Lyapunov Characterization of Robust Stabilization
 Nonlinear Analysis
, 1997
"... One of the fundamental facts in control theory (Artstein's theorem) is the equivalence, for systems affine in controls, between continuous feedback stabilizability to an equilibrium and the existence of smooth control Lyapunov functions. This equivalence breaks down for general nonlinear system ..."
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Cited by 29 (4 self)
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One of the fundamental facts in control theory (Artstein's theorem) is the equivalence, for systems affine in controls, between continuous feedback stabilizability to an equilibrium and the existence of smooth control Lyapunov functions. This equivalence breaks down for general nonlinear systems, not affine in controls. One of the main results in this paper establishes that the existence of smooth Lyapunov functions implies the existence of, in general discontinuous, feedback stabilizers which are insensitive (or robust) to small errors in state measurements. Conversely, it is shown that the existence of such stabilizers in turn implies the existence of smooth control Lyapunov functions. Moreover, it is established that, for general nonlinear control systems under persistently acting disturbances, the existence of smooth Lyapunov functions is equivalent to the existence of (in general, discontinuous) feedback stabilizers which are robust with respect to small measurement errors and sma...
Optimal transportation on noncompact manifolds
"... In this work, we show how to obtain for noncompact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type dr, r> 1, where d is the Riemannian dis ..."
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Cited by 23 (7 self)
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In this work, we show how to obtain for noncompact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type dr, r> 1, where d is the Riemannian distance of a complete Riemannian manifold, hold without any curvature restriction. 1
Optimality Conditions For Optimization Problems With Complementarity Constraints
 SIAM J. Optim
, 1999
"... . Optimization problems with complementarity constraints are closely related to optimization problems with variational inequality constraints and bilevel programming problems. In this paper, under mild constraint qualifications, we derive some necessary and su#cient optimality conditions involving t ..."
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Cited by 20 (11 self)
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. Optimization problems with complementarity constraints are closely related to optimization problems with variational inequality constraints and bilevel programming problems. In this paper, under mild constraint qualifications, we derive some necessary and su#cient optimality conditions involving the proximal coderivatives. As an illustration of applications, the result is applied to the bilevel programming problems where the lower level is a parametric linear quadratic problem. Key words. optimization problems, complementarity constraints, optimality conditions, bilevel programming problems, proximal normal cones AMS subject classifications. 49K99, 90C, 90D65 PII. S1052623497321882 1. Introduction. The main purpose of this paper is to derive necessary and su#cient optimality conditions for the optimization problem with complementarity constraints (OPCC) defined as follows: (OPCC) min f(x, y, u) s.t. #u, #(x, y, u)# = 0, u # 0, #(x, y, u) # 0 (1.1) L(x, y, u) = 0, g(x, y, u) ...
Euler Lagrange and Hamiltonian formalisms in dynamic optimization
 Trans. Amer. Math. Soc
"... Abstract. We consider dynamic optimization problems for systems governed by differential inclusions. The main focus is on the structure of and interrelations between necessary optimality conditions stated in terms of Euler– Lagrange and Hamiltonian formalisms. The principal new results are: an exte ..."
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Cited by 14 (1 self)
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Abstract. We consider dynamic optimization problems for systems governed by differential inclusions. The main focus is on the structure of and interrelations between necessary optimality conditions stated in terms of Euler– Lagrange and Hamiltonian formalisms. The principal new results are: an extension of the recently discovered form of the Euler–Weierstrass condition to nonconvex valued differential inclusions, and a new Hamiltonian condition for convex valued inclusions. In both cases additional attention was given to weakening Lipschitz type requirements on the set–valued mapping. The central role of the Euler type condition is emphasized by showing that both the new Hamiltonian condition and the most general form of the Pontriagin maximum principle for equality constrained control systems are consequences of the Euler–Weierstrass condition. An example is given demonstrating that the new Hamiltonian condition is strictly stronger than the previously known one. 1.
Lipschitzian regularity of the minimizing trajectories for nonlinear optimal control problems
 IN THE JOURNAL MATHEMATICS OF CONTROL, SIGNALS, AND SYSTEMS (MCSS
, 2002
"... We consider the Lagrange problem of optimal control with unrestricted controls and address the question: under what conditions we can assure optimal controls are bounded? This question is related to the one of Lipschitzian regularity of optimal trajectories, and the answer to it is crucial for closi ..."
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Cited by 9 (5 self)
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We consider the Lagrange problem of optimal control with unrestricted controls and address the question: under what conditions we can assure optimal controls are bounded? This question is related to the one of Lipschitzian regularity of optimal trajectories, and the answer to it is crucial for closing the gap between the conditions arising in the existence theory and necessary optimality conditions. Rewriting the Lagrange problem in a parametric form, we obtain a relation between the applicability conditions of the Pontryagin maximum principle to the later problem and the Lipschitzian regularity conditions for the original problem. Under the standard hypotheses of coercivity of the existence theory, the conditions imply that the optimal controls are essentially bounded, assuring the applicability of the classical necessary optimality conditions like the Pontryagin maximum principle. The result extends previous Lipschitzian regularity results to cover
Dynamic Managerial Compensation: a Mechanism Design Approach
, 2009
"... We characterize the optimal incentive scheme for a manager who faces costly e¤ort decisions and whose ability to generate pro…ts for the …rm varies stochastically over time. The optimal contract is obtained as the solution to a dynamic mechanism design problem with hidden actions and persistent shoc ..."
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Cited by 7 (3 self)
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We characterize the optimal incentive scheme for a manager who faces costly e¤ort decisions and whose ability to generate pro…ts for the …rm varies stochastically over time. The optimal contract is obtained as the solution to a dynamic mechanism design problem with hidden actions and persistent shocks to the agent’s productivity. When the agent is riskneutral, the optimal contract can often be implemented with a simple pay package that is linear in the …rm’s pro…ts. Furthermore, the power of the incentive scheme typically increases over time, thus providing a possible justi…cation for the frequent practice of putting more stocks and options in the package of managers with a longer tenure in the …rm. In contrast to other explanations proposed in the literature (e.g., declining disutility of e¤ort or career concerns), the optimality of senioritybased reward schemes is not driven by variations in the agent’s preferences or in his outside option. It results from an optimal allocation of the manager’s informational rents over time. Building on the insights from the riskneutral case, we then explore the properties of optimal incentive schemes for riskaverse managers. We …nd that, other things equal, riskaversion reduces the bene…t of inducing higher e¤ort over time. Whether (riskaverse) managers with a longer tenure receive more or less highpowered incentives than younger ones then depends on the interaction between the degree of risk aversion and the dynamics of the impulse responses for the shocks to the manager’s type. JEL classi…cation: D82
A calculus of epiderivatives applicable to optimization
, 1992
"... When an optimization problem is represented by its essential objective function, which incorporates constraints through infinite penalties, first and secondorder conditions for optimality can be stated in terms of the first and secondorder epiderivatives of that function. Such derivatives also ..."
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Cited by 6 (4 self)
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When an optimization problem is represented by its essential objective function, which incorporates constraints through infinite penalties, first and secondorder conditions for optimality can be stated in terms of the first and secondorder epiderivatives of that function. Such derivatives also are the key to the formulation of subproblems determining the response of a problem’s solution when the data values on which the problem depends are perturbed. It is vital for such reasons to have available a calculus of epiderivatives. This paper builds on a central case already understood, where the essential objective function is the composite of a convex function and a smooth mapping with certain qualifications, in order to develop differentiation rules covering operations such as addition of functions and a more general form of composition. Classes of “amenable” functions are introduced to mark out territory in which this sharper form of nonsmooth analysis can be carried out.
On Nonconvex Subdifferential Calculus in Banach Spaces
, 1995
"... this paper we establish some useful calculus rules in the general Banach space setting. ..."
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Cited by 4 (0 self)
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this paper we establish some useful calculus rules in the general Banach space setting.
Discontinuous Solutions Of The HamiltonJacobi Equations For Exit Time Problems
 SIAM J. Control Optim
, 2000
"... . In general, the value function associated with an exit time problem is a discontinuous function. We prove that the lower (upper) semicontinuous envelope of the value function is a supersolution (subsolution) of the HamiltonJacobi equation involving the proximal subdifferentials (superdifferential ..."
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Cited by 2 (0 self)
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. In general, the value function associated with an exit time problem is a discontinuous function. We prove that the lower (upper) semicontinuous envelope of the value function is a supersolution (subsolution) of the HamiltonJacobi equation involving the proximal subdifferentials (superdifferentials) with subdifferential type (superdifferential type) mixed boundary condition. We also show that if the value function is upper semicontinuous then it is the maximum subsolution of the HamiltonJacobi equation involving the proximal superdifferentials with the natural boundary condition, and if the value function is lower semicontinuous then it is the minimum solution of the HamiltonJacobi equation involving the proximal subdifferentials with a natural boundary condition. Futhermore, if a compatibility condition is satisfied, then the value function is the unique lower semicontinuous solution of the HamiltonJacobi equation with a natural boundary condition and a subdifferential type bound...