Global Convergence of a New Nonmonotone Filter Method for Equality Constrained Optimization

A new nonmonotone filter trust region method is introduced for solving optimization problems with equality constraints. This method directly uses the dominated area of the filter as an acceptability criterion for trial points and allows the dominated area decreasing nonmonotonically. Compared with the filter-type method, our method has more flexible criteria and can avoid Maratos effect in a certain degree. Under reasonable assumptions, we prove that the given algorithm is globally convergent to a first order stationary point for all possible choices of the starting point. Numerical tests are presented to show the effectiveness of the proposed algorithm

A penalty-function-free line search SQP method for nonlinear programming

AbstractWe propose a penalty-function-free non-monotone line search method for nonlinear optimization problems with equality and inequality constraints. This method yields global convergence without using a penalty function or a filter. Each step is required to satisfy a decrease condition for the constraint violation, as well as that for the objective function under some reasonable conditions. The proposed mechanism for accepting steps also combines the non-monotone technique on the decrease condition for the constraint violation, which leads to flexibility and an acceptance behavior comparable with filter based methods. Furthermore, it is shown that the proposed method can avoid the Maratos effect if the search directions are improved by second-order corrections (SOC). So locally superlinear convergence is achieved. We also present some numerical results which confirm the robustness and efficiency of our approach

Hybrid Filter Methods for Nonlinear Optimization

Globalization strategies used by algorithms to solve nonlinear constrained optimization problems must balance the oftentimes conflicting goals of reducing the objective function and satisfying the constraints. The use of merit functions and filters are two such popular strategies, both of which have their strengths and weaknesses. In particular, traditional filter methods require the use of a restoration phase that is designed to reduce infeasibility while ignoring the objective function. For this reason, there is often a significant decrease in performance when restoration is triggered. In Chapter 3, we present a new filter method that addresses this main weakness of traditional filter methods. Specifically, we present a hybrid filter method that avoids a traditional restoration phase and instead employs a penalty mode that is built upon the l-1 penalty function; the penalty mode is entered when an iterate decreases both the penalty function and the constraint violation. Moreover, the algorithm uses the same search direction computation procedure during every iteration and uses local feasibility estimates that emerge during this procedure to define a new, improved, and adaptive margin (envelope) of the filter. Since we use the penalty function (a combination of the objective function and constraint violation) to define the search direction, our algorithm never ignores the objective function, a property that is not shared by traditional filter methods. Our algorithm thusly draws upon the strengths of both filter and penalty methods to form a novel hybrid approach that is robust and efficient. In particular, under common assumptions, we prove global convergence of our algorithm. In Chapter 4, we present a nonmonotonic variant of the algorithm in Chapter 3. For this version of our method, we prove that it generates iterates that converge to a first-order solution from an arbitrary starting point, with a superlinear rate of convergence. We also present numerical results that validate the efficiency of our method. Finally, in Chapter 5, we present a numerical study on the application of a recently developed bound-constrained quadratic optimization algorithm on the dual formulation of sparse large-scale strictly convex quadratic problems. Such problems are of particular interest since they arise as subproblems during every iteration of our new filter methods

Practical implementation of an interior point nonmonotone line search filter method

Versão não definitiva do artigoHere we present a primal-dual interior point nonmonotone line search filter method for nonlinear programming. The filter relies on three measures, the feasibility, the centrality and the optimality presented in the optimality conditions, considers relaxed acceptability criteria for the step size and includes a feasibility restoration phase. The evaluation of the method is until now made on small problems and a comparison is provided with a merit function approach

A globally convergent primal-dual interior-point filter method for nonlinear programming

In this paper, the filter technique of Fletcher and Leyffer (1997) is used to globalize the primal-dual interior-point algorithm for nonlinear programming, avoiding the use of merit functions and the updating of penalty parameters. The new algorithm decomposes the primal-dual step obtained from the perturbed first-order necessary conditions into a normal and a tangential step, whose sizes are controlled by a trust-region type parameter. Each entry in the filter is a pair of coordinates: one resulting from feasibility and centrality, and associated with the normal step; the other resulting from optimality (complementarity and duality), and related with the tangential step. Global convergence to first-order critical points is proved for the new primal-dual interior-point filter algorithm

A Filter Algorithm with Inexact Line Search

A filter algorithm with inexact line search is proposed for solving nonlinear programming problems. The filter is constructed by employing the norm of the gradient of the Lagrangian function to the infeasibility measure. Transition to superlinear local convergence is showed for the proposed filter algorithm without second-order correction. Under mild conditions, the global convergence can also be derived. Numerical experiments show the efficiency of the algorithm