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The main goals of this paper are to: i) relate two iteration-complexity bounds derived for the Mizuno-Todd-Ye predictor-corrector (MTY P-C) algorithm for linear programming (LP), and; ii) study the geometrical structure of the LP central path. The first iteration-complexity bound for the MTY P-C algorithm considered in this paper is expressed in terms of the integral of a certain curvature function over the traversed portion of the central path. The second iteration-complexity bound, derived recently by the authors using the notion of crossover events introduced by Vavasis and Ye, is expressed in terms of a scale-invariant condition number associated with m × n constraint matrix of the LP. In this paper, we establish a relationship between these bounds by showing that the first one can be majorized by the second one. We also establish a geometric result about the central path which gives a rigorous justification based on the curvature of the central path of a claim made by Vavasis and Ye, in view of the behavior of their layered least squares path following LP method, that the central path consists of $${\mathcal{O}}(n^2)$$ long but straight continuous parts while the remaining curved part is relatively “short”. [ABSTRACT FROM AUTHOR]

This paper addresses the local convergence properties of the affine-scaling interior-point algorithm for nonlinear programming. The analysis of local convergence is developed in terms of parameters that control the interior-point scheme and the size of the residual of the linear system that provides the step direction. The analysis follows the classical theory for quasi-Newton methods and addresses q-linear, q-superlinear, and q-quadratic rates of convergence. [ABSTRACT FROM AUTHOR]

In this paper, we propose an augmented Lagrangian affine scaling (ALAS) algorithm for general nonlinear programming, for which a quadratic approximation to the augmented Lagrangian is minimized at each iteration. Different from the classical sequential quadratic programming (SQP), the linearization of nonlinear constraints is put into the penalty term of this quadratic approximation, which results in smooth objective of the subproblem and avoids possible inconsistency among the linearized constraints and trust region constraint. By applying affine scaling techniques to handle the strict bound constraints, an active set type affine scaling trust region subproblem is proposed. Through special strategies of updating Lagrange multipliers and adjusting penalty parameters, this subproblem is able to reduce the objective function value and feasibility errors in an adaptive well-balanced way. Global convergence of the ALAS algorithm is established under mild assumptions. Furthermore, boundedness of the penalty parameter is analyzed under certain conditions. Preliminary numerical experiments of ALAS are performed for solving a set of general constrained nonlinear optimization problems in CUTEr problem library. [ABSTRACT FROM AUTHOR]

This paper presents extensions and further analytical properties of algorithms for linear programming based only on primal scaling and projected gradients of a potential function. The paper contains extensions and analysis of two polynomial-time algorithms for linear programming. We first present an extension of Gonzaga's O( nL) iteration algorithm, that computes dual variables and does not assume a known optimal objective function value. This algorithm uses only affine scaling, and is based on computing the projected gradient of the potential function where x is the vector of primal variables and s is the vector of dual slack variables, and q = n + $$\sqrt n $$ . The algorithm takes either a primal step or recomputes dual variables at each iteration. We next present an alternate form of Ye's O( $$\sqrt n $$ L) iteration algorithm, that is an extension of the first algorithm of the paper, but uses the potential function where q = n + $$\sqrt n $$ . We use this alternate form of Ye's algorithm to show that Ye's algorithm is optimal with respect to the choice of the parameter q in the following sense. Suppose that q = n + n where t⩾0. Then the algorithm will solve the linear program in O( n L) iterations, where r = max{ t, 1 − t}. Thus the value of t that minimizes the complexity bound is t = 1/2, yielding Ye's O( $$\sqrt n $$ L) iteration bound. [ABSTRACT FROM AUTHOR]

In this paper, a continuous method approach is adopted to study both the entire process and the limiting behaviors of the dual affine scaling continuous trajectories for linear programming. Our approach is different from the method presented by Adler and Monteiro (Adler and Monteiro, Math. Program. 50:29-51, ). Many new theoretical results on the trajectories resulting from the dual affine scaling continuous method model for linear programming are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new trust region affine scaling method for nonlinear programming with simple bounds. Our new method is an interior-point trust region method with a new scaling technique. The scaling matrix depends on the distances of the current iterate to the boundaries, the gradient of the objective function and the trust region radius. This scaling technique is different from the existing ones. It is motivated by our analysis of the linear programming case. The trial step is obtained by minimizing the quadratic approximation to the objective function in the scaled trust region. It is proved that our algorithm guarantees that at least one accumulation point of the iterates is a stationary point. Preliminary numerical experience on problems with simple bounds from the CUTEr collection is also reported. The numerical performance reveals that our method is effective and competitive with the famous algorithm LANCELOT. It also indicates that the new scaling technique is very effective and might be a good alternative to that used in the subroutinefminconfrom Matlab optimization toolbox. [ABSTRACT FROM PUBLISHER]

Chen and Zhang [ Sci. China, Ser. A, 45, 1390-1397 (2002)] introduced an affine scaling trust region algorithm for linearly constrained optimization and analyzed its global convergence. In this paper, we derive a new affine scaling trust region algorithm with dwindling filter for linearly constrained optimization. Different from Chen and Zhang's work, the trial points generated by the new algorithm are accepted if they improve the objective function or improve the first order necessary optimality conditions. Under mild conditions, we discuss both the global and local convergence of the new algorithm. Preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]

In this paper we introduce a primal-dual affine scaling method. The method uses a search-direction obtained by minimizing the duality gap over a linearly transformed conic section. This direction neither coincides with known primal-dual affine scaling directions (Jansen et al., 1993; Monteiro et al., 1990), nor does it fit in the generic primal-dual method (Kojima et al., 1989). The new method requires $$O(\sqrt n L)$$ main iterations. It is shown that the iterates follow the primal-dual central path in a neighbourhood larger than the conventional $$\mathcal{N}_2 $$ neighbourhood. The proximity to the primal-dual central path is measured by trigonometric functions. [ABSTRACT FROM AUTHOR]

The conic model proposed by Davidon (1980) [Davidon, WC (1980). Conic approximations and collinear scalings for optimizers. SIAM Journal on Numerical Analysis, 17, 268-281.] is an extension of quadratic model incorporating more information than quadratic model. In this paper, we propose a conic affine scaling dogleg method for bound constrained optimization. Under some mild conditions, the sequence generated by our algorithm converges to a stationary point. Numerical tests on a series of standard testing problems confirm the efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper we present a scaling-invariant, interior-point, predictor-corrector type algorithm for linear programming (LP) whose iteration-complexity is polynomially bounded by the dimension and the logarithm of a certain condition number of the LP constraint matrix. At the predictor stage, the algorithm either takes the step along the standard affine scaling (AS) direction or a new trust-region type direction, whose construction depends on a scaling-invariant bipartition of the variables determined by the AS direction. This contrasts with the layered least squares direction introduced in S. Vavasis and Y. Ye [Math. Program., 74 (1996), pp. 79-120], whose construction depends on multiple-layered partitions of the variables that are not scaling-invariant. Moreover, it is shown that the overall arithmetic complexity of the algorithm (weakly) depends on the right-hand side and the cost of the LP in view of the work involved in the computation of the trust region steps. [ABSTRACT FROM AUTHOR]

In this paper we present a new iteration-complexity bound for the Mizuno--Todd--Ye predictor-corrector (MTY P-C) primal-dual interior-point algorithm for linear programming. The analysis of the paper is based on the important notion of crossover events introduced by Vavasis and Ye. For a standard form linear program min {cTχ: Aχ = b, &chi ≥ 0 } with decision variable ..., we show that the MTY P-C algorithm, started from a well-centered interior-feasible solution with duality gap nμ0 finds an interior-feasible solution with duality gap less than nn in O (T(μ0/η)+n3.5 log (Χ*A)) iterations, where T(t ) =min {n² log (log t), log t} for all t>0 and Χ*A is a scaling invariant condition number associated with the matrix A. More specifically,Χ*A is the infimum of all the conditions numbers XAD, where D varies over the set of positive diagonal matrices. Under the setting of the Turing machine model, our analysis yields an O (n3.5 LA + min {n² log L, L}) iteration-complexity bound for the MTY P-C algorithm to find a primal-dual optimal solution, where LA and L are the input sizes of the matrix A and the data (A,b,c ), respectively. This contrasts well with the classical iteration-complexity bound for the MTY P-C algorithm, which depends linearly on L instead of log L. [ABSTRACT FROM AUTHOR]

A new trust-region and affine scaling algorithm for linearly constrained optimization is presented in this paper. Under no nondegenerate assumption, we prove that any limit point of the sequence generated by the new algorithm satisfies the first order necessary condition and there exists at least one limit point of the sequence which satisfies the second order necessary condition. Some preliminary numerical experiments are reported. [ABSTRACT FROM AUTHOR]

In this paper we show that the primal-dual Dikin affine scaling algorithm for linear programming of Jansen. Roos and Terlaky enhances an asymptotical $$O(\sqrt n L)$$ complexity by using corrector steps. We also show that the result remains valid when the method is applied to positive semi-definite linear complementarity problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an affine scaling cubic regularization algorithm for solving optimization problem without available derivatives subject to bound constraints employing a polynomial interpolation approach to handle the unavailable derivatives of the original objective function. We first define an affine scaling cubic model of the approximate objective function which is obtained by the polynomial interpolation approach with an affine scaling method. At each iteration a candidate search direction is determined by solving the affine scaling cubic regularization subproblem and the new iteration is strictly feasible by way of an interior backtracking technique. The global convergence and local superlinear convergence of the proposed algorithm are established under some mild conditions. Preliminary numerical results are reported to show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a derivative-free affine scaling inexact Levenberg–Marquardt method with interior backtracking line search technique is considered for solving linear inequality constrained nonlinear systems. The proposed algorithm is designed to take advantage of the problem structure by building polynomial interpolation models for each function of nonlinear systems subject to the linear inequality constraints on variables. Each iterate switches to backtracking step generated by affine scaling inexact Levenberg–Marquardt method and satisfies strict interior point feasibility by line search backtracking technique. Under local error bounded assumption, the method is superlinear and quadratic convergent on F ( x ). The results of numerical experiments are reported to show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an affine scaling optimal path approach in association with nonmonotonic interior backtracking line search technique for nonlinear optimization subject to linear constraints. We shall employ the eigensystem decomposition and affine scaling mapping to form affine scaling optimal curvilinear path very easily. By using interior backtracking line search technique, each iterate switches to trial step of strict interior feasibility. The nonmonotone criterion is used to speed up the convergence progress in the contours of objective function with large curvature. Theoretical analysis is given which prove that the proposed algorithm is globally convergent and has a local superlinear convergence rate under some reasonable conditions. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm. [Copyright &y& Elsevier]

In this paper we propose an affine scaling interior algorithm via conjugate gradient path for solving nonlinear equality systems subject to bounds on variables. By employing the affine scaling conjugate gradient path search strategy, we obtain an iterative direction by solving the linearize model. By using the line search technique, we will find an acceptable trial step length along this direction which is strictly feasible and makes the objective function nonmonotonically decreasing. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. Furthermore, the numerical results of the proposed algorithm indicate to be effective. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we propose an affine scaling interior algorithm via Lanczos path for solving nonlinear equality systems subject to bounds on variables. Employing the affine scaling Lanczos path search strategy, we obtain an iterative direction by solving the linearize model. By using the line search backtracking technique, we will find an acceptable trial step length along this direction which is strictly feasible and makes the objective function nonmonotonically decreasing. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. Furthermore, the numerical results of the proposed algorithm indicate to be effective. [Copyright &y& Elsevier]

Abstract: This paper presents an affine scaling reduced preconditional conjugate gradient path approach in association with nonmonotonic interior backtracking line search technique for the linear constrained optimization. Employing the affine scaling preconditional conjugate gradient to form the curvilinear path and using interior backtracking line search technique, each iterate switches to trial step of strict interior feasibility. The nonmonotone criterion is used to speed up the convergence progress in the contours of objective function with large curvature. Theoretical analysis are given which prove that the proposed algorithm is globally convergent and has a local superlinear convergence rate under some reasonable conditions. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new interior point algorithm for linear programming by combining the affine scaling algorithm and the primal path-following algorithm. We define a series of subproblems whose feasible region contracts toward the set of optimal solutions. In each iteration, we take a step in the steepest descent direction in a transformed space to reduce the objective value. Our preliminary computational results are very favorable. [Copyright &y& Elsevier]

Abstract: In this paper we propose a nonmonotonic interior point backtracking strategy to modify the reduced projective affine scaling trust region algorithm for solving minimum optimization subject to both nonlinear equality and linear inequality constraints. The general full trust region subproblem for solving the minimum optimization is decomposed to a pair of trust region subproblems in horizontal and vertical subspaces of linearize equality constraints and extended affine scaling equality constraints by QR decomposition of an affine scaling matrix and an orthonormal basis on the null subspace. The horizontal subproblem in the proposed algorithm is defined by minimizing a quadratic projective reduced Hessian function subject only to an ellipsoidal trust region constraint, while the vertical subproblem is also defined by the least squares subproblem subject only to an ellipsoidal trust region constraint. Combining trust region strategy with line search technique will switch to strictly feasible interior point step generated by a component direction of the two trust region subproblems. By adopting the l 1 penalty function as the merit function, the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. The second-order correction step and a nonmonotonic criterion are used to overcome Maratos effect and speed up the convergence progress in some ill-conditioned cases, respectively. [Copyright &y& Elsevier]

In this paper, we propose a new nonmonotonic interior point backtracking strategy to modify the reduced projective affine scaling trust region algorithm for solving optimization subject to nonlinear equality and linear inequality constraints. The general full trust region subproblem for solving the nonlinear equality and linear inequality constrained optimization is decomposed to a pair of trust region subproblems in horizontal and vertical subspaces of linearize equality constraints and extended affine scaling equality constraints. The horizontal subproblem in the proposed algorithm is defined by minimizing a quadratic projective reduced Hessian function subject only to an ellipsoidal trust region constraint in a null subspace of the tangential space, while the vertical subproblem is also defined by the least squares subproblem subject only to an ellipsoidal trust region constraint. By introducing the Fletcher''s penalty function as the merit function, trust region strategy with interior point backtracking technique will switch to strictly feasible interior point step generated by a component direction of the two trust region subproblems. The global convergence of the proposed algorithm while maintaining fast local convergence rate of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion should bring about speeding up the convergence progress in some high nonlinear function conditioned cases. [Copyright &y& Elsevier]

In this paper we propose a new interior affine scaling region algorithm with nonmonotonic interior point backtracking technique for nonlinear optimization subject to linear equality and inequality constraints. The trust region subproblem in the proposed algorithm is defined by minimizing a quadratic function subject only to an affine scaling ellipsoidal constraint in a null subspace of the extended equality constraints. Using both trust region strategy and line search technique, the affine scaling trust region subproblem at each iteration generates backtracking interior step to obtain a new accepted step. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases. [Copyright &y& Elsevier]

This note derives bounds on the length of the primal-dual affinescaling directions associated with a linearly constrained convexprogram satisfying the following conditions: 1) the problem has asolution satisfying strict complementarity, 2) the Hessian of theobjective function satisfies a certain invariance property. Weillustrate the usefulness of these bounds by establishing thesuperlinear convergence of the algorithm presented in Wright andRalph [22] for solving the optimality conditions associatedwith a linearly constrained convex program satisfying the aboveconditions. [ABSTRACT FROM AUTHOR]

A primal, interior point method is developed for linear programming problems for which the linear objective function is to be maximised over polyhedra that are not necessarily in standard form. This algorithm concurs with the affine scaling method of Dikin when the polyhedron is in standard form, and satisfies the usual conditions imposed for using that method. If the search direction is regarded as a function of the current iterate, then it is shown that this function has a unique, continuous extension to the boundary. In fact, on any given face, this extension is just the value the search direction would have for the problem of maximising the objective function over that face. This extension is exploited to prove convergence. The algorithm presented here can be used to exploit such special constraint structure as bounds, ranges, and free variables without increasing the size of the linear programming problem. [ABSTRACT FROM AUTHOR]

Two primal-dual affine scaling algorithms for linear programming are extended to semidefinite programming. The algorithms do not require (nearly) centered starting solutions, and can be initiated with any primal-dual feasible solution. The first algorithm is the Dikin-type affine scaling method of Jansen et al. (1993b) and the second the classical affine scaling method of Monteiro et al. (1990). The extension of the former has a worst-case complexity bound of O(τ0nL) iterations, where τ0 is a measure of centrality of the the starting solution, and the latter a bound of O(τ0nL2) iterations. [ABSTRACT FROM AUTHOR]

Abstract This paper focuses on a class of nonlinear optimization subject to linear inequality constraints with unavailable-derivative objective functions. We propose a derivative-free trust-region methods with interior backtracking technique for this optimization. The proposed algorithm has four properties. Firstly, the derivative-free strategy is applied to reduce the algorithm’s requirement for first- or second-order derivatives information. Secondly, an interior backtracking technique ensures not only to reduce the number of iterations for solving trust-region subproblem but also the global convergence to standard stationary points. Thirdly, the local convergence rate is analyzed under some reasonable assumptions. Finally, numerical experiments demonstrate that the new algorithm is effective.

We propose an inexact affine scaling Levenberg-Marquardt method for solving bound-constrained semismooth equations under the local error bound assumption which is much weaker than the standard nonsingularity condition. The affine scaling Levenberg-Marquardt equation is based on a minimization of the squared Euclidean norm of linearized model adding a quadratic affine scaling matrix to find a solution which belongs to the bounded constraints on variable. The global convergence and the superlinear convergence rate are proved. Numerical results show that the new algorithm is efficient. [ABSTRACT FROM AUTHOR]

Within the framework of affine-scaling trust-region methods for bound-constrained problems, we discuss the use of an inexact dogleg method as a tool for simultaneously handling the trust-region and the bound constraints while seeking for an approximate minimizer of the model. Then, we focus on large-scale bound-constrained systems of nonlinear equations which often arise in practical applications when some of the unknowns are naturally subject to constraints due to physical arguments. We introduce an inexact affine-scaling method for such a class of problems that employs the inexact dogleg procedure. Global convergence results are established without any Lipschitz assumption on the Jacobian matrix, and locally fast convergence is shown under standard assumptions. Convergence analysis is performed without specifying the scaling matrix that is used to handle the bounds, and a rather general class of scaling matrices is allowed in actual algorithms. Numerical results showing the performance of the method are also given. [ABSTRACT FROM AUTHOR]

Recently an affine scaling, interior point algorithm ASL was developed for box constrained optimization problems with a single linear constraint (Gonzalez-Lima et al., SIAM J. Optim. 21:361-390, ). This note extends the algorithm to handle more general polyhedral constraints. With a line search, the resulting algorithm ASP maintains the global and R-linear convergence properties of ASL. In addition, it is shown that the unit step version of the algorithm (without line search) is locally R-linearly convergent at a nondegenerate local minimizer where the second-order sufficient optimality conditions hold. For a quadratic objective function, a sublinear convergence property is obtained without assuming either nondegeneracy or the second-order sufficient optimality conditions. [ABSTRACT FROM AUTHOR]

We introduce an inexact Gauss-Newton trust-region method for solving bound-constrained nonlinear least-squares problems where, at each iteration, a trust-region subproblem is approximately solved by the Conjugate Gradient method. Provided a suitable control on the accuracy to which we attempt to solve the subproblems, we prove that the method has global and asymptotic fast convergence properties. Some numerical illustration is also presented. [ABSTRACT FROM AUTHOR]

We study a new trust region affine scaling method for general bound constrained optimization problems. At each iteration, we compute two trial steps. We compute one along some direction obtained by solving an appropriate quadratic model in an ellipsoidal region. This region is defined by an affine scaling technique. It depends on both the distances of current iterate to boundaries and the trust region radius. For convergence and avoiding iterations trapped around nonstationary points, an auxiliary step is defined along some newly defined approximate projected gradient. By choosing the one which achieves more reduction of the quadratic model from the two above steps as the trial step to generate next iterate, we prove that the iterates generated by the new algorithm are not bounded away from stationary points. And also assuming that the second-order sufficient condition holds at some nondegenerate stationary point, we prove the Q-linear convergence of the objective function values. Preliminary numerical experience for problems with bound constraints from the CUTEr collection is also reported. [ABSTRACT FROM AUTHOR]

We propose a first-order interior-point method for linearly constrained smooth optimization that unifies and extends first-order affine-scaling method and replicator dynamics method for standard quadratic programming. Global convergence and, in the case of quadratic program, (sub)linear convergence rate and iterate convergence results are derived. Numerical experience on simplex constrained problems with 1000 variables is reported. [ABSTRACT FROM AUTHOR]

A new class of infeasible interior point methods for solving sufficient linear complementarity problems (LCPs) requiring one matrix factorization and m backsolves at each iteration is proposed and analyzed. The algorithms from this class use a large (N−∞ ) neighborhood of an infeasible central path associated with the complementarity problem and an initial positive, but not necessarily feasible, starting point. The Q-order of convergence of the complementarity gap, the residual, and the iteration sequence is m+ 1 for problems that admit a strict complementarity solution and (m+1)/2 for general sufficient LCPs. The methods do not depend on the handicap κ of the sufficient LCP. If the starting point is feasible (or “almost” feasible), the proposed algorithms have O((1 + κ)(1 +log m√1 + κ )√n L) iteration complexity, while if the starting point is “large enough,” the iteration complexity is O((1 + κ)2+1/m(1 + log m√1 +κ )n L). [ABSTRACT FROM AUTHOR]

Based on a differentiable merit function proposed by Taji et al. in “Math. Prog. Stud., 58, 1993, 369–383”, the authors propose an affine scaling interior trust region strategy via optimal path to modify Newton method for the strictly monotone variational inequality problem subject to linear equality and inequality constraints. By using the eigensystem decomposition and affine scaling mapping, the authors from an affine scaling optimal curvilinear path very easily in order to approximately solve the trust region subproblem. Theoretical analysis is given which shows that the proposed algorithm is globally convergent and has a local quadratic convergence rate under some reasonable conditions. [ABSTRACT FROM AUTHOR]

The authors propose an affine scaling modified gradient path method in association with reduced projective Hessian and nonmonotonic interior backtracking line search techniques for solving the linear equality constrained optimization subject to bounds on variables. By employing the QR decomposition of the constraint matrix and the eigensystem decomposition of reduced projective Hessian matrix in the subproblem, the authors form affine scaling modified gradient curvilinear path very easily. By using interior backtracking line search technique, each iterate switches to trial step of strict interior feasibility. The global convergence and fast local superlinear/quadratical convergence rates of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

A first order affine scaling method and two mth order affine scaling methods for solving monotone linear complementarity problems (LCPs) are presented. All three methods produce iterates in a wide neighborhood of the central path. The first order method has O(nL²(log nL²) (log log nL²)) iteration complexity. If the LCP admits a strict complementary solution, then both the duality gap and the iteration sequence converge superlinearly with Q-order two. If m = Ω(log(√nL)), then both higher order methods have O(√n)L iteration complexity. The Q-order of convergence of one of the methods is (m + 1) for problems that admit a strict complementarity solution, while the Q-order of convergence of the other method is (m + 1)/2 for general monotone LCPs. [ABSTRACT FROM AUTHOR]

An algorithm for the solution of a semismooth system of equations with box constraints is described. The method is an affine-scaling trust-region method. All iterates generated by this method are strictly feasible. In this way, possible domain violations outside or on the boundary of the box are avoided. The method is shown to have strong global and local convergence properties under suitable assumptions, in particular, when the method is used with a special scaling matrix. Numerical results are presented for a number of problems arising from different areas. [ABSTRACT FROM AUTHOR]

A class of new affine-scaling interior-point Newton-type methods are considered for the solution of optimization problems with bound constraints. The methods are shown to be locally quadratically convergent under the strong second order sufficiency condition without assuming strict complementarity of the solution. The new methods differ from previous ones by Coleman and Li [Mathematical Programming, 67 (1994), pp. 189-224] and Heinkenschloss, Ulbrich, and Ulbrich [Mathematical Programming, 86 (1999), pp. 615-635] mainly in the choice of the scaling matrix. The scaling matrices used here have stronger smoothness properties and allow the application of standard results from nonsmooth analysis in order to obtain a relatively short and elegant local convergence result. An important tool for the definition of the new scaling matrices is the correct identification of the degenerate indices. Some illustrative numerical results with a comparison of the different scaling techniques are also included. [ABSTRACT FROM AUTHOR]

Consider solving a linear program in standard form where the constraint matrix A is m x n, with n ≫ m ≫ 1. Such problems arise, for example, as the result of finely discretizing a semi-infinite program. The cost per iteration of typical primal-dual interior-point methods on such problems is O(m²n). We propose to reduce that cost by replacing the normal equation matrix, AD²AT, where D is a diagonal matrix, with a "reduced" version (of same dimension), AQDQ²AQT, where Q is an index set including the indices of M most nearly active (or most violated) dual constraints at the current iterate, with M ≥ m a prescribed integer. This can result in a speedup of close to n/¦Q¦ at each iteration. Promising numerical results are reported for constraint-reduced versions of a dual-feasible affine-scaling algorithm and of Mehrotra's predictor-corrector method [S. Mehrotra, SIAM J. Optim., 2 (1992), pp. 575-601]. In particular, while it could be expected that neglecting a large portion of the constraints, especially at early iterations, may result in a significant deterioration of the search direction, it appears that the total number of iterations typically remains essentially constant as the size of the reduced constraint set is decreased down to some threshold. In some cases this threshold is a small fraction of the total set. In the case of the affine-scaling algorithm, global convergence and local quadratic convergence are proved. [ABSTRACT FROM AUTHOR]