Abstract
The aggregate constraint homotopy method uses a single smoothing constraint instead ofm-constraints to reduce the dimension of its homotopy map, and hence it is expected to be more efficient than the combined homotopy interior point method when the number of constraints is very large. However, the gradient and Hessian of the aggregate constraint function are complicated combinations of gradients and Hessians of all constraint functions, and hence they are expensive to calculate when the number of constraint functions is very large. In order to improve the performance of the aggregate constraint homotopy method for solving nonlinear programming problems, with few variables and many nonlinear constraints, a flattened aggregate constraint homotopy method, that can save much computation of gradients and Hessians of constraint functions, is presented. Under some similar conditions for other homotopy methods, existence and convergence of a smooth homotopy path are proven. A numerical procedure is given to implement the proposed homotopy method, preliminary computational results show its performance, and it is also competitive with the state-of-the-art solver KNITRO for solving large-scale nonlinear optimization.
Highlights
IntroductionWe consider the following nonlinear programming problem: min f (x) , (1)
In this paper, we consider the following nonlinear programming problem: min f (x), (1)s.t. g (x) ≤ 0, where x ∈ f : Rn →Rn is the variable, g(x) R, and gi(x) : Rn → R,= i (g1(x), . . . , gm(x))T, = 1, . . . , m, are three times continuously differentiable, and m is very large, but n is moderate
It is known as exponential penalty function. By using it for all constraint functions of problem (1), an aggregate constraint homotopy method was presented by Yu et al in [19], whose global convergence was obtained under the condition that the feasible set satisfies the weak normal cone condition
Summary
We consider the following nonlinear programming problem: min f (x) , (1). In [10, 11], Feng et al proposed a homotopy method for nonlinear programming (1), which was called the combined homotopy interior point (abbreviated by CHIP) method; Abstract and Applied Analysis its global convergence was proven under the normal cone condition (see below for its definition) for the feasible set as well as some common conditions. It is known as exponential penalty function (see [18]) By using it for all constraint functions of problem (1), an aggregate constraint homotopy (abbreviated by ACH) method was presented by Yu et al in [19], whose global convergence was obtained under the condition that the feasible set satisfies the weak normal cone condition.
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