A penalty-free-type nonmonotone trust-region method for nonlinear constrained optimization

Abstract

In this paper, we propose a penalty-free-type nonmonotone trust region method for solving general nonlinear programming problem. The algorithmic framework yields global convergence without using any penalty function. We analyze the global convergence of the main algorithm for the degenerate problems and also give the globally convergent results under the linear independence constraint qualification. The preliminary numerical tests are reported.

Introduction

Consider the following nonlinear programming problem:

$$\begin{array}{cc}\hfill & \min f(x)\hfill \\ \hfill & \mathrm{s}\text{.}\mathrm{t}\text{.}c(x)=0\text{,}\hfill \end{array}$$

$$l⩽x⩽u\text{,}$$

where x ∈ Rⁿ, c(x) = (c₁(x), c₂(x), … , c_m(x))^T, m < n, f(x) and c_i(x) (i = 1, 2, … , m) are real functions defined in D = {x ∈ Rⁿ∣l ⩽ x ⩽ u}. We assume that l_i < u_i (i = 1, 2, … , n) and they may be infinite.

Let g(x) denote the gradient of f(x) and A(x) = (∇c₁(x), ∇c₂(x), … , ∇c_m(x)). Throughout this paper, we let ∥ · ∥ and ∥ · ∥_∞ be the l₂-norm and the l_∞-norm on Rⁿ, respectively. We also use f_k for f(x_k), c_k for c(x_k), etc.

Now, we define the substationary point and ϕ-stationary point of (1.1) as follows.

Definition 1.1 [17]

For some x* ∈ D, if there exist μ_l ⩾ 0, μ_u ⩾ 0 such that

$$\begin{array}{cc}\hfill & A(x^{\ast })c(x^{\ast })-{\mu }_l+{\mu }_u=0\text{,}\hfill \\ \hfill & {\mu }_l^{\mathrm{T}}(x^{\ast }-l)=0\text{,}{\mu }_u^{\mathrm{T}}(u-x^{\ast })=0\text{,}\hfill \end{array}$$

then x* is called a ϕ-stationary point of (1.1).

Definition 1.2 [5]

For some x* ∈ D, if there exist μ_l ⩾ 0, μ_u ⩾ 0 and λ ∈ R^m such that

$$\begin{array}{cc}\hfill & g(x^{\ast })-A(x^{\ast })\lambda -{\mu }_l+{\mu }_u=0\text{,}\hfill \\ \hfill & {\mu }_l^{\mathrm{T}}(x^{\ast }-l)=0\text{,}{\mu }_u^{\mathrm{T}}(u-x^{\ast })=0\text{,}\hfill \\ \hfill & l⩽x^{\ast }⩽u\text{,}\hfill \end{array}$$

then x* is called a substationary point of (1.1). Moreover, if c(x*) = 0, then x* is called a stationary point of (1.1).

Given x_k ∈ D, an estimate of the solution, the problem (1.1) is often solved by the sequential quadratic programming (SQP) method and it is always assumed that a search direction d_k can be computed by solving the following quadratic programming subproblem:

$$\begin{array}{cc}\hfill & \min g_k^{\mathrm{T}}d+\frac{1}{2}{d}^{\mathrm{T}}{B}_{k}d\hfill \\ \hfill & \mathrm{s}\text{.}\mathrm{t}\text{.}{c}_k+A_k^{\mathrm{T}}d=0\text{,}\hfill \end{array}$$

$$l⩽x_k+d⩽u\text{,}$$

where B_k ∈ R^n×n is a symmetric matrix. And then the new iterate point is x_k+1 = x_k + α_{k dk}, where α_k > 0 is a step length and α_k depends on some line search technique [23]. Here it is usually assumed that B_k is positive definite and the linearized constraints are consistent.

There are many trust region methods for equality constrained nonlinear programming. For example, Byrd, Schnabel and Shultz [3], Celis, Dennis and Tapia [4], El Alem [14], Powell and Yuan [22], Vardi [26], Dennis and Vicente [13]. Several extensions of trust region method to the problems involving inequality constraints have been proposed. Some of them are based on trust region methods for box-constrained problems, see, e.g., Byrd [1], Omojokun [20], Dennis et al. [11], Conn et al. [8], Dennis and Vicente [13], Chen et al. [6]. There are other related approaches which combine trust region method with interior point method, such as Dennis et al. [12], Plantenga [21],Vicente [27], Byrd et al. [2], Cleman and Li [7]. For SQP filter methods, the global convergence has been established in Fletcher et al. [15], [16]. Filter methods and its predecessor do not use any penalty function, either. There are still other methods for solving (1.1), e.g., the generalized reduced gradient in Lasdon [19], the augmented Lagrangian approach in Chen et al. [5], Conn et al. [9], Gomes et al. [17] and so on.

In this paper, we will use [5], [17] and [25] as our main references. However, there are several related approaches and recent extensions that should be mentioned. We first refer to a class of trust region algorithms introduced by Byrd [1], Omojokun [20], Dennis et al. [11], the filtering idea introduced by Fletcher and Leyffer [15], [16]. Ulbrich et al. [25] presents a nonmonotone trust region methods for nonlinear equality constrained optimization without a penalty function. Chen et al. [5] and Gomes et al. [17] use trust-region method with the augmented Lagrangian function. The main contributions of our paper is to generalize the idea and the methods above from with nonlinear equality constraints or only with box-constraints to with general nonlinear constrained optimization problem. The method does not require a restoration procedure and does not use any penalty function. Under no linear independence constraint qualification, using the concept of substationary point [5] and ϕ-stationary point [17], we give the analysis of the global convergence. We also prove that there exists at least a limit point of the sequence generated by the new algorithm, which is a stationary point of the original problem, under the linear independence constraint qualification.

This paper is organized as follows. In Section 2, the formal algorithm is described. In Section 3, we prove that, under mild conditions, this algorithm is well defined and some global convergence results with or without linear independence constraint qualification are given. Some numerical experiments for problems from [18] and [24] collection are reported in Section 4. Finally, we give some final remarks on this approach.