2,831 research outputs found
An augmented Lagrangian interior-point method using directions of negative curvature
The original publication is available at www.springerlink.comWe describe an efficient implementation of an interior-point algorithm for non-convex problems
that uses directions of negative curvature. These directions should ensure convergence to second-order KKT
points and improve the computational efficiency of the procedure. Some relevant aspects of the implementation
are the strategy to combine a direction of negative curvature and a modified Newton direction, and
the conditions to ensure feasibility of the iterates with respect to the simple bounds. The use of multivariate
barrier and penalty parameters is also discussed, as well as the update rules for these parameters.We analyze
the convergence of the procedure; both the linesearch and the update rule for the barrier parameter behave
appropriately. As the main goal of the paper is the practical usage of negative curvature, a set of numerical
results on small test problems is presented. Based on these results, the relevance of using directions of negative
curvature is discussed.Research supported by Spanish MEC grant TIC2000-1750-C06-04; Research supported by Spanish MEC grant BEC2000-0167Publicad
An augmented lagrangian interior point method using diretions of negative curvature
We describe an efficient implementation of an interior-point algorithm for non-convex problems that uses directions of negative curvature. These directions should ensure convergence to second-order KKT points and improve the computational efficiency of the procedure. Some relevant aspects of the implementation are the strategy to combine a direction of negative curvature and a modified Newton direction, and the conditions to ensure feasibility of the iterates with respect to the simple bounds. The use of multivariate barrier and penalty parameters is also discussed, as well as the update rules for these parameters. Finally, numerical results on a set oftest problems are presented
Combining search directions using gradient flows
The efficient combination of directions is a significant problem in line search methods that either use negative curvature. or wish to include additional information such as the gradient or different approximations to the Newton direction. In thls paper we describe a new procedure to combine several of these directions within an interior-point primal-dual algorithm. Basically. we combine in an efficient manner a modified Newton direction with the gradient of a merit function and a direction of negative curvature. is it exists. We also show that the procedure is well-defined. and it has reasonable theoretical properties regarding the convergence of the method. We also present numerical results from an implementation of the proposed algorithm on a set of small test problems from the CUTE collection
An interior-point method for mpecs based on strictly feasible relaxations.
An interior-point method for solving mathematical programs with equilibrium constraints (MPECs) is proposed. At each iteration of the algorithm, a single primaldual step is computed from each subproblem of a sequence. Each subproblem is defined as a relaxation of the MPEC with a nonempty strictly feasible region. In contrast to previous approaches, the proposed relaxation scheme preserves the nonempty strict feasibility of each subproblem even in the limit. Local and superlinear convergence of the algorithm is proved even with a less restrictive strict complementarity condition than the standard one. Moreover, mechanisms for inducing global convergence in practice are proposed. Numerical results on the MacMPEC test problem set demonstrate the fast-local convergence properties of the algorithm
AN INTERIOR-POINT METHOD FOR MPECs BASED ON STRICTLY FEASIBLE RELAXATIONS.
An interior-point method for solving mathematical programs with equilibrium constraints (MPECs) is proposed. At each iteration of the algorithm, a single primaldual step is computed from each subproblem of a sequence. Each subproblem is defined as a relaxation of the MPEC with a nonempty strictly feasible region. In contrast to previous approaches, the proposed relaxation scheme preserves the nonempty strict feasibility of each subproblem even in the limit. Local and superlinear convergence of the algorithm is proved even with a less restrictive strict complementarity condition than the standard one. Moreover, mechanisms for inducing global convergence in practice are proposed. Numerical results on the MacMPEC test problem set demonstrate the fast-local convergence properties of the algorithm.
An Alternating Trust Region Algorithm for Distributed Linearly Constrained Nonlinear Programs, Application to the AC Optimal Power Flow
A novel trust region method for solving linearly constrained nonlinear
programs is presented. The proposed technique is amenable to a distributed
implementation, as its salient ingredient is an alternating projected gradient
sweep in place of the Cauchy point computation. It is proven that the algorithm
yields a sequence that globally converges to a critical point. As a result of
some changes to the standard trust region method, namely a proximal
regularisation of the trust region subproblem, it is shown that the local
convergence rate is linear with an arbitrarily small ratio. Thus, convergence
is locally almost superlinear, under standard regularity assumptions. The
proposed method is successfully applied to compute local solutions to
alternating current optimal power flow problems in transmission and
distribution networks. Moreover, the new mechanism for computing a Cauchy point
compares favourably against the standard projected search as for its activity
detection properties
Combining search directions using gradient flows
The original publication is available at www.springerlink.comThe efficient combination of directions is a significant problem in line search methods that either use negative curvature, or wish to include additional information such as the gradient or different approximations
to the Newton direction.
In this paper we describe a new procedure to combine several of these directions within an interior-point primal-dual algorithm. Basically, we combine in an efficient manner a modified Newton direction with the
gradient of a merit function and a direction of negative curvature, if it exists.We also show that the procedure
is well-defined, and it has reasonable theoretical properties regarding the rate of convergence of the method.
We also present numerical results from an implementation of the proposed algorithm on a set of small
test problems from the CUTE collection.Research supported by Spanish MEC grants BEC2000-0167 and PB98-0728Publicad
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