Strong Convergence on the Aggregate Constraint-Shifting HomotopyMethod for Solving General Nonconvex Programming

In the paper, the aggregate constraint-shifting homotopy method for solving general nonconvex nonlinear programming is considered. +e aggregation is only about inequality constraint functions. Without any cone condition for the constraint functions, the existence and convergence of the globally convergent solution to the K-K-T system are obtained for both feasible and infeasible starting points under much weaker conditions.

It is well known that the solution of the optimization problem can be obtained through solving the K-K-T system of the convex nonlinear problem, but for the nonconvex nonlinear problem, we can only obtain the solution to the K-K-T system of problem (1). Homotopy method has been paid much attention as an important globally convergent computational method in finding solutions to various nonlinear problems since it was introduced and studied by Kellogg et al. [1], Smale [2], and Chow et al. [3]. However, the original homotopy is only single homotopy and needs much strong assumptions when solving nonlinear problems. In the 1990s, a combined homotopy interior point (CHIP) method was firstly proposed for solving nonconvex programming under the normal cone condition by Feng and Yu in [4]. From then on, various CHIP methods, as an efficiently implementable algorithm, were widely used and newly constructed for solving general nonconvex programming, fixed point problems, complementarity problems, variational inequality, and so on, see, e.g., [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].
In 2001, for reducing the dimension of the systems arising in the numerically tracing process and weakening convergent conditions, Yu et al. [21] proposed an aggregate constraint homotopy method (ACH method) for nonconvex programming by using the so-called aggregate function of the constraints. In 2018, a new ACH method for nonlinear programming problems with inequality and equality constraints was presented in [22]. However, the ACH method still belongs to CHIP since it requires the initial point which was also in the original feasible set. In 2006, to avoid the disadvantage of CHIP must choose the initial point in the feasible set, a constraint-shifting combined homotopy infeasible interior-point method in which the initial point can be chosen in both feasible and infeasible sets for solving nonlinear programming with only inequality constraints was proposed by Yu and Shang in [23,24]. In 2012, to extend the constraint-shifting combined homotopy method to solve the general nonlinear programming, another new combined homotopy infeasible interior-point method for solving nonconvex programming with both inequality and equality constraints was proposed in [25], in which only inequality constraints need to satisfy the normal cone condition. From then on, more constraint-shifting homotopy equations were constructed and extended for solving nonlinear programming, principal-agent problem, fixed point problem, and so on, see, e.g., [26][27][28][29][30]. However, these combined homotopy methods usually required some cone conditions for proving the strong convergence of the existence of the smooth homotopy pathway.
By the enlightenment of the above references, without any cone condition, an aggregate constraint combined homotopy infeasible interior-point method for solving nonconvex nonlinear programming with both inequality and equality constraints is constructed, and the global convergence under much weaker conditions is obtained in the paper. e remainder of this paper is organized as follows. In Section 2, the homotopy equation is constructed, and some lemmas from differential topology are introduced. In Section 3, the main results will be presented, and the existence and convergence of a smooth path from any given point in the infeasible set to the solution of K-K-T systems are proved. In Section 4, the numerical algorithm is presented.
When μ � 0, homotopy equation (11) turns to the K-K-T system When μ � 1, homotopy equation (11), H(w, w 0 , 1) � 0, has a unique simple solution e following lemmas from differential topology will be used in the next section. At first, let U ⊂ R n be an open set, and let ϕ: U ⟶ R p be a C α (α > max 0, n − p ) mapping; we say that y ∈ R p is a regular value for ϕ if Lemma 4 (see [31]). Let V ⊂ R n and U ⊂ R m be open sets, and let ϕ: If 0 ∈ R k is a regular value of ϕ, then for almost all a ∈ V, 0 is a regular value of ϕ a � F(a, ·).
Lemma 6 (see [31]). A one-dimensional smooth manifold is diffeomorphic to a unit circle or a unit interval.
If ‖(1 − μ k )y k ‖ ⟶ ∞, the discussion is the same as the following case (ii).