Curvilinear path and trust region in unconstrained optimization: A convergence analysis

Abstract

In this paper we propose a general algorithm for solving unconstrained optimization problems. The basic step of the algorithm consists in finding a "good" successor point to the current iterate by choosing it along a curvilinear path and within a trust region. This scheme is due to Powell and it has been applied by Sorensen to a particular type of path. We give a series of properties that an arbitrary path should satisfy in order to achieve global convergence and fast asymptotic convergence. We review various paths that have been proposed in the literature and study the extent to which they satisfy our properties.

An affine scaling interior trust region method via optimal path for solving monotone variational inequality problem with linear constraints

“…It should be noted that Γ 2 (t 2 (τ )) is defined only when M k is indefinite and g i k = 0 for all i ∈ {1, 2, · · · , n− l} with φ i = φ 1 < 0, which is referred to hard case (see [2]) for unconstrained optimization, and for other cases, Γ(τ ) is defined only for 0 ≤ τ < 1 T , that is, Γ(τ ) = Γ 1 (t 1 (τ )).…”

“…In order to avoid solving the trust region subproblem frequently, we pay no attention to the trust region constraint and go to search for the trial step at each iteration along some curvilinear path. By using the idea of the optimal path of general trust region subproblem in [2], we now form an approximate affine scaling interior optimal path of trust region subproblem (S k ).…”

Nonmonotonic Reduced Projected Hessian Method Via an Affine Scaling Interior Modified Gradient Path for Bounded-Constrained Optimization

“…In this section, we form the modified gradient path as proposed by Bulteau and Vial for solving the unconstrained optimization in [3] (also see [9]). When the parameter τ varies in the interval [0, +∞), the solution points form the curvilinear path and emanate from the origin point x 0 .…”

“…By employing the QR decomposition of the linear equality constraint matrix in order to overcome the difficulty of the feasible constraints in trust region subproblem, the subproblem in the proposed algorithm is a general trust region subproblem defined by minimizing a quadratic function only subject to an ellipsoid constraint in the null subspace. On the other hand, in order to avoid the difficulties of the strictly feasible constraints in affine scaling trust region subproblem, a special affine scaling curve, called affine scaling interior modified gradient path, which can be expressed by the eigenvalues and eigenvectors of the reduced projective Hessian, was suggested in using general trust region strategy by Bulteau and Vial [3] . It is helpful to include a scaling matrix for the variables to adjust to interior feasibility and reduced projective Hessian for a null space of the affine scaling linear equality constraints.…”

“…An attractive idea is to solve the trust region subproblem approximately along a curve originating at x k within the trust region. The various curvilinear path trust region algorithms are available (see [3][4][5], for example). The trust region strategy in association with line search suggested in [6] motivates to switch to the line search technique by employing the backtracking steps at trial step which may be unaccepted in trust region strategy, since the trial step should provide a direction of sufficient descent.…”