A globally and superlinearly convergent trust region method for LC 1 optimization problems

Abstract. A new trust region algorithm for solving convex LC1 optimization problem is present-ed. It is proved that the algorithm is globally convergent and the rate of convergence is superlin-ear under some reasonable assumptions.     

A globally and superlinearly convergent modified SQP-filter method

In this paper, we presented a modified SQP-filter method based on the modified quadratic subproblem proposed by Zhou (J. Global Optim. 11, 193–2005, 1997). In contrast with the SQP methods, each iteration this algorithm only needs to solve one quadratic programming subproblems and it is always feasible. Moreover, it has no demand on the initial point. With the filter technique, the algorithm shows good numerical results. Under some conditions, the globally and superlinearly convergent properties are given.     

A superlinearly and quadratically convergent SQP type feasible method for constrained optimization

A new SQP type feasible method for inequality constrained optimization is presented, it is a combination of a master algorithm and an auxiliary algorithm which is taken only in finite iterations. The directions of the master algorithm are generated by only one quadratic programming, and its step-size is always one, the directions of the auxiliary algorithm are new “secondorder“ feasible descent. Under suitable assumptions, the algorithm is proved to possess global and strong convergence, superlinear and quadratic convergence.     

A superlinearly convergent method to linearly constrained optimization problems under degeneracy

Most papers concerning nonlinear programming problems with linear constraints assume linear independence of the gradients of the active constraints at any feasible point. In this paper we remove this assumption and give an algorithm and prove its convergency. Also, under appropriate assumptions on the objective function, including one which could be viewed as an extension of the strict complementary slackness condition at the optimal solution, we prove the rate of convergence of the algorithm to be superlinear.     

A globally and superlinearly convergent quasi-Newton method for general box constrained variational inequalities without smoothing approximation

A new quasi-Newton algorithm for the solution of general box constrained variational inequality problem (GVI(l, u, F, f)) is proposed in this paper. It is based on a reformulation of the variational inequality problem as a nonsmooth system of equations by using the median operator. Without smoothing approximation, the proposed quasi-Newton algorithm is directly applied to solve this class of nonsmooth equations. Under appropriate assumptions, it is proved that the algorithmic sequence globally and superlinearly converges to a solution of the equation reformulation and also of GVI(l, u, F, f). Numerical results show that our new algorithm works quite well.     

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.Mathematics Subject Classification (1991): 65K05, 90C06, 90C29, 90C30Support for this author was provided by CRPC grant CCR–9120008.Support for this author was provided by CRPC grant CCR–9120008.Support for this author was provided by Centro de Matemática da Universidade de Coimbra, by FCT under grant POCTI/35059/MAT/2000, by the European Union under grant IST-2000-26063, and by Fundaç\ ao Calouste Gulbenkian. The author would also like to thank the IBM T.J. Watson Research Center and the Institute for Mathematics and Its Applications for their local support.     

Superlinear and quadratic convergence of some primal-dual interior point methods for constrained optimization

This paper proves local convergence rates of primal-dual interior point methods for general nonlinearly constrained optimization problems. Conditions to be satisfied at a solution are those given by the usual Jacobian uniqueness conditions. Proofs about convergence rates are given for three kinds of step size rules. They are: (i) the step size rule adopted by Zhang et al. in their convergence analysis of a primal-dual interior point method for linear programs, in which they used single step size for primal and dual variables; (ii) the step size rule used in the software package OB1, which uses different step sizes for primal and dual variables; and (iii) the step size rule used by Yamashita for his globally convergent primal-dual interior point method for general constrained optimization problems, which also uses different step sizes for primal and dual variables. Conditions to the barrier parameter and parameters in step size rules are given for each case. For these step size rules, local and quadratic convergence of the Newton method and local and superlinear convergence of the quasi-Newton method are proved. A preliminary version of this paper was presented at the conference “Optimization-Models and Algorithms” held at the Institute of Statistical Mathematics, Tokyo, March 1993.     

A superlinearly convergent ODE-type trust region algorithm for nonsmooth nonlinear equations

This paper presents a new trust region algorithm for solving nonsmooth nonlinear equation problems which posses the smooth plus non-smooth decomposition. At each iteration, this method obtains a trial step by solving a system of linear equations, hence avoiding the need for solving a quadratic programming subproblem with a trust region bound. From a computational point of view, this approach may reduce computational effort and hence improve computational efficiency. Furthermore, it is proved under appropriate assumptions that this algorithm is globally and locally super-linearly convergent. Some numerical examples are reported.     

A globally convergent trust region algorithm for optimization with general constraints and simple bounds

1.IntroductionInthispaper,weconsiderthefollowingnonlinearprogr~ngproblemwherec(x)=(c,(x),c2(2),',We(.))',i(x)andci(x)(i=1,2,',m)arerealfunctions*ThisworkissupPOrtedbytheNationalNaturalScienceFOundationofChinaandtheManagement,DecisionandinformationSystemLab,theChineseAcademyofSciences.definedinD={xEReIISx5u}.Weassumethath相似文献   

A primal-dual trust region algorithm for nonlinear optimization

This paper concerns general (nonconvex) nonlinear optimization when first and second derivatives of the objective and constraint functions are available. The proposed method is based on finding an approximate solution of a sequence of unconstrained subproblems parameterized by a scalar parameter. The objective function of each unconstrained subproblem is an augmented penalty-barrier function that involves both primal and dual variables. Each subproblem is solved using a second-derivative Newton-type method that employs a combined trust region and line search strategy to ensure global convergence. It is shown that the trust-region step can be computed by factorizing a sequence of systems with diagonally-modified primal-dual structure, where the inertia of these systems can be determined without recourse to a special factorization method. This has the benefit that off-the-shelf linear system software can be used at all times, allowing the straightforward extension to large-scale problems. Numerical results are given for problems in the COPS test collection.Mathematics Subject Classification (2000): 49M37, 65F05, 65K05, 90C30This paper is dedicated to Roger Fletcher on the occasion of his 65th birthday     

A specialized primal-dual interior point method for the plastic truss layout optimization

We are concerned with solving linear programming problems arising in the plastic truss layout optimization. We follow the ground structure approach with all possible connections between the nodal points. For very dense ground structures, the solutions of such problems converge to the so-called generalized Michell trusses. Clearly, solving the problems for large nodal densities can be computationally prohibitive due to the resulting huge size of the optimization problems. A technique called member adding that has correspondence to column generation is used to produce a sequence of smaller sub-problems that ultimately approximate the original problem. Although these sub-problems are significantly smaller than the full formulation, they still remain large and require computationally efficient solution techniques. In this article, we present a special purpose primal-dual interior point method tuned to such problems. It exploits the algebraic structure of the problems to reduce the normal equations originating from the algorithm to much smaller linear equation systems. Moreover, these systems are solved using iterative methods. Finally, due to high degree of similarity among the sub-problems after preforming few member adding iterations, the method uses a warm-start strategy and achieves convergence within fewer interior point iterations. The efficiency and robustness of the method are demonstrated with several numerical experiments.     

A globally convergent interior point algorithm for non-convex nonlinear programming

In this paper, a new algorithm for tracing the combined homotopy path of the non-convex nonlinear programming problem is proposed. The algorithm is based on the techniques of β$\beta$-cone neighborhood and a combined homotopy interior point method. The residual control criteria, which ensures that the obtained iterative points are interior points, is given by the condition that ensures the β$\beta$-cone neighborhood to be included in the interior part of the feasible region. The global convergence and polynomial complexity are established under some hypotheses.     

A superlinearly convergent method of quasi-strongly sub-feasible directions with active set identifying for constrained optimization

Combining the norm-relaxed sequential quadratic programming (SQP) method and the idea of method of quasi-strongly sub-feasible directions (MQSSFD) with active set identification technique, a new SQP algorithm for solving nonlinear inequality constrained optimization is proposed. Unlike the previous work, at each iteration of the proposed algorithm, the norm-relaxed quadratic programming (QP) subproblem only consists of the constraints corresponding to an active identification set. Moreover, the high-order correction direction (used to avoid the Maratos effect) is yielded by solving a system of linear equations (SLE) which also includes only the constraints and their gradients corresponding to the active identification set, therefore, the scale and the computation cost of the high-order correction directions are further decreased. The arc search in our algorithm can effectively combine the initialization processes with the optimization processes, and the iteration points can get into the feasible set after a finite number of iterations. Furthermore, the arc search conditions are weaker than the previous work, and the computation cost is further reduced. The global convergence is proved under the Mangasarian–Fromovitz constraint qualification (MFCQ). If the strong second-order sufficient conditions are satisfied, then the active constraints are exactly identified by the identification set. Without the strict complementarity, superlinear convergence can be obtained. Finally, some elementary numerical results are reported.     

A ROBUST SUPERLINEARLY CONVERGENT ALGORITHM FOR LINEARLY CONSTRAINED OPTIMIZATION PROBLEMS UNDER DEGENERACY

1.IntroductionTheproblemconsideredinthispaperiswhereX={xER"laTx5hi,jEI={l,.'.,m}},ajeR"(jEI)areallcolumn*ThisresearchissupportedbytheNationalNaturalSciencesFoundationofChinaandNaturalSciencesFoundationofHunanProvince.vectors,hiERI(j6I)areallscalars,andf:R"-- Risacontinuouslydifferentiablefunction.Weonlyconsiderinequalityconstraintsheresinceanyequalitycanbeexpressedastwoinequalities.Withoutassumingregularityofthelinearconstraints,thereisnotanydifficultyinextendingtheresultstothegenera…     

A superlinearly convergent projection method for constrained systems of nonlinear equations

In this paper, a new projection method for solving a system of nonlinear equations with convex constraints is presented. Compared with the existing projection method for solving the problem, the projection region in this new algorithm is modified which makes an optimal stepsize available at each iteration and hence guarantees that the next iterate is more closer to the solution set. Under mild conditions, we show that the method is globally convergent, and if an error bound assumption holds in addition, it is shown to be superlinearly convergent. Preliminary numerical experiments also show that this method is more efficient and promising than the existing projection method. This work was done when Yiju Wang visited Chongqing Normal University.     

An interior trust region algorithm for solving linearly constrained nonlinear optimization

In this paper, an interior point algorithm based on trust region techniques is proposed for solving nonlinear optimization problems with linear equality constraints and nonnegative variables. Unlike those existing interior-point trust region methods, this proposed method does not require that a general quadratic subproblem with a trust region bound be solved at each iteration. Instead, a system of linear equations is solved to get a search direction, and then a linesearch of Armijo type is performed in this direction to obtain a new iteration point. From a computational point of view, this approach may in general reduce a computational effort, and thus improve the computational efficiency. Under suitable conditions, it is proven that any accumulation of the sequence generated by the algorithm satisfies the first-order optimality condition.