Results 1  10
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27
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 Newton Barrier method for Minimizing a Sum of Euclidean Norms subject to linear equality constraints
, 1995
"... An algorithm for minimizing a sum of Euclidean Norms subject to linear equality constraints is described. The algorithm is based on a recently developed Newton barrier method for the unconstrained minimization of a sum of Euclidean norms (MSN ). The linear equality constraints are handled using an e ..."
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Cited by 19 (2 self)
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An algorithm for minimizing a sum of Euclidean Norms subject to linear equality constraints is described. The algorithm is based on a recently developed Newton barrier method for the unconstrained minimization of a sum of Euclidean norms (MSN ). The linear equality constraints are handled using an exact L 1 penalty function which is made smooth in the same way as the Euclidean norms. It is shown that the dual problem is to maximize a linear objective function subject to homogeneous linear equality constraints and quadratic inequalities. Hence the suggested method also solves such problems efficiently. In fact such a problem from plastic collapse analysis motivated this work. Numerical results are presented for large sparse problems, demonstrating the extreme efficiency of the method. Keywords: Sum of Norms, Nonsmooth Optimization, Duality, Newton Barrier Method. AMS(MOS) subject classification: 65K05, 90C06, 90C25, 90C90. Abbreviated title: A Newton barrier method. Supported by the ...
Pseudonormality and a Lagrange Multiplier Theory for Constrained Optimization
, 2000
"... We consider optimization problems with equality, inequality, and abstract set constraints, and we explore various characteristics of the constraint set that imply the existence of Lagrange multipliers. We prove a generalized version of the FritzJohn theorem, and we introduce new and general conditi ..."
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Cited by 12 (2 self)
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We consider optimization problems with equality, inequality, and abstract set constraints, and we explore various characteristics of the constraint set that imply the existence of Lagrange multipliers. We prove a generalized version of the FritzJohn theorem, and we introduce new and general conditions that extend and unify the major constraint qualifications. Among these conditions, two new properties, pseudonormality and quasinormality, emerge as central within the taxonomy of interesting constraint characteristics. In the case where there is no abstract set constraint, these properties provide the connecting link between the classical constraint qualifications and two distinct pathways to the existence of Lagrange multipliers: one involving the notion of quasiregularity and Farkas' Lemma, and the other involving the use of exact penalty functions. The second pathway also applies in the general case where there is an abstract set constraint.
Steering Exact Penalty Methods for Nonlinear Programming
, 2007
"... This paper reviews, extends and analyzes a new class of penalty methods for nonlinear optimization. These methods adjust the penalty parameter dynamically; by controlling the degree of linear feasibility achieved at every iteration, they promote balanced progress toward optimality and feasibility. I ..."
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Cited by 10 (0 self)
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This paper reviews, extends and analyzes a new class of penalty methods for nonlinear optimization. These methods adjust the penalty parameter dynamically; by controlling the degree of linear feasibility achieved at every iteration, they promote balanced progress toward optimality and feasibility. In contrast with classical approaches, the choice of the penalty parameter ceases to be a heuristic and is determined, instead, by a subproblem with clearly defined objectives. The new penalty update strategy is presented in the context of sequential quadratic programming (SQP) and sequential linearquadratic programming (SLQP) methods that use trust regions to promote convergence. The paper concludes with a discussion of penalty parameters for merit functions used in line search methods.
A Second Derivative SQP Method: Local Convergence 30 Practical Issues
 SIAM Journal of Optimization
"... results for a secondderivative SQP method for minimizing the exact ℓ1merit function for a fixed value of the penalty parameter. To establish this result, we used the properties of the socalled Cauchy step, which was itself computed from the socalled predictor step. In addition, we allowed for th ..."
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Cited by 6 (0 self)
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results for a secondderivative SQP method for minimizing the exact ℓ1merit function for a fixed value of the penalty parameter. To establish this result, we used the properties of the socalled Cauchy step, which was itself computed from the socalled predictor step. In addition, we allowed for the computation of a variety of (optional) SQP steps that were intended to improve the efficiency of the algorithm. Although we established global convergence of the algorithm, we did not discuss certain aspects that are critical when developing software capable of solving general optimization problems. In particular, we must have strategies for updating the penalty parameter and better techniques for defining the positivedefinite matrix Bk used in computing the predictor step. In this paper we address both of these issues. We consider two techniques for defining the positivedefinite matrix Bk—a simple diagonal approximation and a more sophisticated limitedmemory BFGS update. We also analyze a strategy for updating the penalty parameter based on approximately minimizing the ℓ1penalty function over a sequence of increasing values of the penalty parameter. Algorithms based on exact penalty functions have certain desirable properties. To be practical, however, these algorithms must be guaranteed to avoid the socalled Maratos effect. We show that a nonmonotone variant of our algorithm avoids this phenomenon and, therefore, results in asymptotically superlinear local convergence; this is verified by preliminary numerical results on the Hock and Shittkowski test set. Key words. Nonlinear programming, nonlinear inequality constraints, sequential quadratic programming, ℓ1penalty function, nonsmooth optimization AMS subject classifications. 49J52, 49M37, 65F22, 65K05, 90C26, 90C30, 90C55 1. Introduction. In [19]
Infeasibility Detection and SQP Methods for Nonlinear Optimization
, 2008
"... This paper addresses the need for nonlinear programming algorithms that provide fast local convergence guarantees no matter if a problem is feasible or infeasible. We present an activeset sequential quadratic programming method derived from an exact penalty approach that adjusts the penalty paramet ..."
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Cited by 4 (2 self)
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This paper addresses the need for nonlinear programming algorithms that provide fast local convergence guarantees no matter if a problem is feasible or infeasible. We present an activeset sequential quadratic programming method derived from an exact penalty approach that adjusts the penalty parameter appropriately to emphasize optimality over feasibility, or vice versa. Conditions are presented under which superlinear convergence is achieved in the infeasible case. Numerical experiments illustrate the practical behavior of the method.
On the Convergence of Successive Linear Programming Algorithms
, 2003
"... We analyze the global convergence properties of a class of penalty methods for nonlinear programming. These methods include successive linear programming approaches, and more speci cally the SLPEQP approach presented in [1]. Every iteration requires the solution of two trust region subproblems inv ..."
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Cited by 4 (1 self)
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We analyze the global convergence properties of a class of penalty methods for nonlinear programming. These methods include successive linear programming approaches, and more speci cally the SLPEQP approach presented in [1]. Every iteration requires the solution of two trust region subproblems involving linear and quadratic models, respectively. The interaction between the trust regions of these subproblems requires careful consideration. It is shown under mild assumptions that there exist an accumulation point which is a critical point for the penalty function.
Constrained LAV State Estimation Using PenaltyFunctions
 IEEE Transactions on Power Systems
, 1997
"... Inequality constraints are often needed in optimization problems in order to deal with uncertainty. This paper introduces a simple technique that allows enforcement of inequality constraints in l1 norm problems without any modifications to existing programs. The solution of l1 norm problems is requi ..."
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Cited by 4 (0 self)
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Inequality constraints are often needed in optimization problems in order to deal with uncertainty. This paper introduces a simple technique that allows enforcement of inequality constraints in l1 norm problems without any modifications to existing programs. The solution of l1 norm problems is required, for example, in implementing LAV (Least Absolute Value) state estimators in electric power systems. The paper shows how LAV state estimators with inequality constraints can be useful for estimating the state of external systems. This is important in a competitiveenvironment where precise information about a utility's neighboring systems may not be available.
Exact Penalty Methods
 In I. Ciocco (Ed.), Algorithms for Continuous Optimization
, 1994
"... . Exact penalty methods for the solution of constrained optimization problems are based on the construction of a function whose unconstrained minimizing points are also solution of the constrained problem. In the first part of this paper we recall some definitions concerning exactness properties of ..."
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Cited by 3 (1 self)
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. Exact penalty methods for the solution of constrained optimization problems are based on the construction of a function whose unconstrained minimizing points are also solution of the constrained problem. In the first part of this paper we recall some definitions concerning exactness properties of penalty functions, of barrier functions, of augmented Lagrangian functions, and discuss under which assumptions on the constrained problem these properties can be ensured. In the second part of the paper we consider algorithmic aspects of exact penalty methods; in particular we show that, by making use of continuously differentiable functions that possess exactness properties, it is possible to define implementable algorithms that are globally convergent with superlinear convergence rate towards KKT points of the constrained problem. 1 Introduction "It would be a major theoretic breakthrough in nonlinear programming if a simple continuously differentiable function could be exhibited with th...
Enhanced Optimality Conditions and Exact Penalty Functions
 PRODEEDINGS OF ALLERTON CONFERENCE, ALLERTON PARK
, 2000
"... We consider optimization problems with equality, inequality, and abstract set constraints, and we explore various characteristics of the constraint set that imply the existence of Lagrange multipliers. We prove a generalized version of the FritzJohn theorem, and we introduce new and general conditi ..."
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Cited by 3 (0 self)
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We consider optimization problems with equality, inequality, and abstract set constraints, and we explore various characteristics of the constraint set that imply the existence of Lagrange multipliers. We prove a generalized version of the FritzJohn theorem, and we introduce new and general conditions that extend and unify the major constraint qualifications. Among these conditions, a new property, pseudonormality, provides the connecting link between the classical constraint qualifications and the use of exact penalty functions.