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61
Optimization Problems with perturbations, A guided tour
 SIAM REVIEW
, 1996
"... This paper presents an overview of some recent and significant progress in the theory of optimization with perturbations. We put the emphasis on methods based on upper and lower estimates of the value of the perturbed problems. These methods allow to compute expansions of the value function and app ..."
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Cited by 73 (10 self)
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This paper presents an overview of some recent and significant progress in the theory of optimization with perturbations. We put the emphasis on methods based on upper and lower estimates of the value of the perturbed problems. These methods allow to compute expansions of the value function and approximate solutions in situations where the set of Lagrange multipliers may be unbounded, or even empty. We give rather complete results for nonlinear programming problems, and describe some partial extensions of the method to more general problems. We illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal.
Sensitivity Analysis of Optimization Problems Under Second Order Regular Constraints
, 1996
"... We present a perturbation theory for finite dimensional optimization problems subject to abstract constraints satisfying a second order regularity condition. We derive Lipschitz and Holder expansions of approximate optimal solutions, under a directional constraint qualification hypothesis and vari ..."
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Cited by 22 (7 self)
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We present a perturbation theory for finite dimensional optimization problems subject to abstract constraints satisfying a second order regularity condition. We derive Lipschitz and Holder expansions of approximate optimal solutions, under a directional constraint qualification hypothesis and various second order sufficient conditions that take into account the curvature of the set defining the constraints of the problem. We also show how the theory applies to semidefinite optimization and, more generally, to semiinfinite programs in which the contact set is a smooth manifold and the quadratic growth condition in the constraint space holds. As a final application we provide a result on differentiability of metric projections in finite dimensional spaces.
Variational Analysis of Functionals of a Poisson Process
, 1997
"... Let F be a functional of a Poisson process whose distribution is determined by the intensity measure #. Considering the expectation E # F as a function on the cone M of positive finite measures #,we derive closed form expressions for the Frechet derivatives of all orders that generalise the perturb ..."
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Cited by 20 (11 self)
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Let F be a functional of a Poisson process whose distribution is determined by the intensity measure #. Considering the expectation E # F as a function on the cone M of positive finite measures #,we derive closed form expressions for the Frechet derivatives of all orders that generalise the perturbation analysis formulae for Poisson processes. Variational methods developed for the space allow us to obtain first and secondorder sufficient conditions for differenttypes of constrained optimisation problems for E # F . We study in detail optimisation in the class of measures with a fixed total mass a and develop technique that often allows us to obtain the asymptotic behaviour of the optimal intensity measure in the high intensity settings when a grows to in#nity. We give applications of our methods to design of experiments, spline approximation of convex functions, optimal placement of stations in telecommunication studies and others. We scetch possible numerical algorithms of the steepest descend type based on the obtained explicit form of the gradient.
Second Order Sufficient Optimality Conditions For Nonlinear Parabolic Control Problems With State Constraints
 J. FOR ANALYSIS AND ITS APPL
, 1996
"... In this paper, optimal control problems for semilinear parabolic equations with distributed and boundary controls are considered. Pointwise constraints on the control and on the state are given. Main emphasis is laid on the discussion of second order sufficient optimality conditions. Sufficiency for ..."
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Cited by 20 (7 self)
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In this paper, optimal control problems for semilinear parabolic equations with distributed and boundary controls are considered. Pointwise constraints on the control and on the state are given. Main emphasis is laid on the discussion of second order sufficient optimality conditions. Sufficiency for local optimality is verified under different assumptions imposed on the dimension of the domain and on the smoothness of the given data.
Optimization Techniques for Solving Elliptic Control Problems with Control and State Constraints. Part 2: Distributed Control
 Comp. Optim. Applic
"... : Part 2 continues the study of optimization techniques for elliptic control problems subject to control and state constraints and is devoted to distributed control. Boundary conditions are of mixed Dirichlet and Neumann type. Necessary conditions of optimality are formally stated in form of a local ..."
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Cited by 18 (3 self)
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: Part 2 continues the study of optimization techniques for elliptic control problems subject to control and state constraints and is devoted to distributed control. Boundary conditions are of mixed Dirichlet and Neumann type. Necessary conditions of optimality are formally stated in form of a local Pontryagin minimum principle. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. The problems are formulated as AMPL [13] scripts and several optimization codes are applied. In particular, it is shown that a recently developed interior point method is able to solve theses problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for dierent types of controls including bang{bang controls. The necessary conditions of optimality are checked numerically in the presence of active control and state constraints....
Strong Duality and Minimal Representations for Cone Optimization
, 2008
"... The elegant results for strong duality and strict complementarity for linear programming, LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primaldual optimal pair that satisfies strict complementarity. This failure is ..."
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Cited by 14 (2 self)
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The elegant results for strong duality and strict complementarity for linear programming, LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primaldual optimal pair that satisfies strict complementarity. This failure is tied to the nonclosure of sums of nonpolyhedral closed cones. We take a fresh look at known and new results for duality, optimality, constraint qualifications, and strict complementarity, for linear cone optimization problems in finite dimensions. These results include: weakest and universal constraint qualifications, CQs; duality and characterizations of optimality that hold without any CQ; geometry of nice and devious cones; the geometric relationships between zero duality gaps, strict complementarity, and the facial structure of cones; and, the connection between theory and empirical evidence for lack of a CQand failure of strict complementarity. One theme is the notion of minimal representation of the cone and the constraints in order to regularize the problem and avoid both the theoretical and numerical difficulties that arise due to (near) loss of a CQ. We include a discussion on obtaining these representations efficiently.
Steepest descent algorithms in space of measures
 Statistics and Computing
, 2000
"... The paper describes descent type algorithms suitable for solving optimisation problems for functionals that depend on measures. We mention several examples of such problems that appear in optimal design, cluster analysis and optimisation of spatial distribution of coverage processes. ..."
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Cited by 13 (4 self)
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The paper describes descent type algorithms suitable for solving optimisation problems for functionals that depend on measures. We mention several examples of such problems that appear in optimal design, cluster analysis and optimisation of spatial distribution of coverage processes.
Nonsmooth Newtonlike Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces
, 2002
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General Optimality Conditions for Constrained Convex Control Problems
 SIAM Journal On Control and Optimization
, 1996
"... In this paper we investigate some optimal convex control problems, with mixed constraints on the state and the control. We give a general condition which allows to set optimality conditions for non qualified problems (in the Slater sense). Then we give some applications and examples involving genera ..."
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Cited by 11 (7 self)
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In this paper we investigate some optimal convex control problems, with mixed constraints on the state and the control. We give a general condition which allows to set optimality conditions for non qualified problems (in the Slater sense). Then we give some applications and examples involving generalized bangbang results.
CHARACTERIZATIONS OF OPTIMALITY WITHOUT CONSTRAINT QUALIFICATION FOR THE Abstract Convex program
, 1982
"... We consider the general abstract convex program (P) minimize [(x), subject to g(x) E S, where f is an extended convex functional on X, g: X ~ Y is Sconvex, S is a closed convex cone and X and Y are topological linear spaces. We present primal and dual characterizations for (P). These characterizat ..."
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Cited by 11 (6 self)
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We consider the general abstract convex program (P) minimize [(x), subject to g(x) E S, where f is an extended convex functional on X, g: X ~ Y is Sconvex, S is a closed convex cone and X and Y are topological linear spaces. We present primal and dual characterizations for (P). These characterizations are derived by reducing the problem to a standard Lagrange multiplier problem. Examples given include operator constrained problems as well as semiinfinite programming problems.