Combined Homotopy Interior Point Method Research Articles

A noninterior point homotopy method for semi-infinite programming problems

In this paper, we present a new homotopy method which is a non-interior point homotopy method for solving semi-infinite programming problems. Under suitable assumptions, we prove that the method determines a smooth path from a given point. The new homotopy method generalizes the existing combined homotopy interior point method for semi-infinite programming problems to unbounded set, moreover, it is more convenient in that it enlarges the choice scope of the initial point. Some numerical examples are given to show its efficiency.

Homotopy method for a class of multiobjective optimization problems with equilibrium constraints

In this paper, we present a combined homotopy interior point method for solving multiobjective programs with equilibrium constraints. Under suitable conditions, we prove the existence and convergence of a smooth homotopy path from almost any interior point to a solution of the K-K-T system. Numerical results are presented to show the effectiveness of this algorithm.

A spline smoothing homotopy method for nonconvex nonlinear programming

Homotopy methods are globally convergent under weak conditions and robust; however, the efficiency of a homotopy method is closely related with the construction of the homotopy map and the path tracing algorithm. Different homotopies may behave very different in performance even though they are all theoretically convergent. In this paper, a spline smoothing homotopy method for nonconvex nonlinear programming is developed using cubic spline to smooth the max function of the constraints of nonlinear programming. Some properties of spline smoothing function are discussed and the global convergence of spline smoothing homotopy under the weak normal cone condition is proven. The spline smoothing technique uses a smooth constraint instead of m constraints and acts also as an active set technique. So the spline smoothing homotopy method is more efficient than previous homotopy methods like combined homotopy interior point method, aggregate constraint homotopy method and other probability one homotopy methods. Numerical tests with the comparisons to some other methods show that the new method is very efficient for nonlinear programming with large number of complicated constraints.

A constraint shifting homotopy method for finding a minimal efficient solution of nonconvex multiobjective programming

In this paper, for finding a minimal efficient solution of nonconvex multiobjective programming, a constraint shifting combined homotopy is constructed and the global convergence is obtained under some mild conditions. The method requires that the initial point needs to be only in the shifted feasible set not necessarily in the original feasible set, and the normal cone condition need only be satisfied in the boundary of the shifted feasible set not the original constraint set. Some numerical tests are done and made comparison with a combined homotopy interior point method. The results show that our method is feasible and effective.

A homotopy method for nonlinear second-order cone programming

This paper is concerned with a homotopy method for solving nonlinear second-order cone programming problems. The method extends to this setting a combined homotopy interior point method, recently introduced for solving nonlinear programming problems. Global convergence of a smooth curve determined by constructed homotopy is proven under mild conditions. Some numerical results are reported and show that the considered algorithm is applicable and efficient.

The Flattened Aggregate Constraint Homotopy Method for Nonlinear Programming Problems with Many Nonlinear Constraints

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.

A Weaker Constraint Qualification of Globally Convergent Homotopy Method for a Multiobjective Programming Problem

In this paper, we prove that the combined homotopy interior point method for a multiobjective programming problem introduced in Ref. [1] remains valid under a weaker constrained qualification—the Mangasarian-Fromovitz constrained qualification, instead of linear independence constraint qualification. The algorithm generated by this method associated to the Karush-Kuhn-Tucker points of the multiobjective programming problem is proved to be globally convergent.

A constraint shifting homotopy method for general non-linear programming

In this article, a constraint shifting homotopy method (CSHM) is proposed for solving non-linear programming with both equality and inequality constraints. A new homotopy is constructed, and existence and global convergence of a homotopy path determined by it are proven. All problems that can be solved by the combined homotopy interior point method (CHIPM) can also be solved by the proposed method. In contrast to the combined homotopy infeasible interior point method (CHIIPM), it needs a weaker regularity condition. And the starting point in the proposed method is not necessarily a feasible point or an interior point, so it is more convenient to be implemented than CHIPM and CHIIPM. Numerical results show that the proposed algorithm is feasible and effective.

A globally convergent interior point algorithm for non-convex nonlinear programming

In this paper, a new algorithm for tracing the combined homotopy path of the non-convex nonlinear programming problem is proposed. The algorithm is based on the techniques of β -cone neighborhood and a combined homotopy interior point method. The residual control criteria, which ensures that the obtained iterative points are interior points, is given by the condition that ensures the β -cone neighborhood to be included in the interior part of the feasible region. The global convergence and polynomial complexity are established under some hypotheses.

Homotopy Method for a General Multiobjective Programming Problem

In this paper, we present a combined homotopy interior-point method for a general multiobjective programming problem. The algorithm generated by this method associated to Karush–Kuhn–Tucker points of the multiobjective programming problem is proved to be globally convergent under some basic assumptions.

Solving the Karush–Kuhn–Tucker system of a nonconvex programming problem on an unbounded set

In the papers [G.C. Feng, B. Yu, Combined homotopy interior point method for nonlinear programming problems, in: H. Fujita, M. Yamaguti (Eds.), Advances in Numerical Mathematics; Proceedings of the Second Japan–China Seminar on Numerical Mathematics, in: Lecture Notes in Numerical and Applied Analysis, vol. 14, Kinokuniya, Tokyo, 1995, pp. 9–16; G.C. Feng, Z.H. Lin, B. Yu, Existence of an interior pathway to a Karush–Kuhn–Tucker point of a nonconvex programming problem, Nonlinear Analysis 32 (1998) 761–768; Z.H. Lin, B. Yu, G.C. Feng, A combined homotopy interior point method for convex programming problem, Applied Mathematics and Computation 84 (1997) 193–211], a combined homotopy interior method was presented and global convergence results obtained for nonconvex nonlinear programming when the feasible set is bounded and satisfies the so called normal cone condition. However, for when the feasible set is not bounded, no result has so far been obtained. In this paper, a combined homotopy interior method for nonconvex programming problems on the unbounded feasible set is considered. Under suitable additional assumptions, boundedness of the homotopy path, and hence global convergence, is proven.

A polynomial path following algorithm for convex programming

In this paper, based on combined homotopy interior point method we propose an interior point algorithm for convex nonlinear programming. The algorithm ensures that the obtained iterative points are interior points of the feasible set in terms of the technique of β-cone neighborhood. We establish the global convergence of the algorithm. Furthermore, it is shown that the algorithm has O ( n L ) iteration complexity. The preliminary numerical experiments indicate that the algorithm is efficient.

Modified homotopy methods to solve fixed points of self-mapping in a broader class of nonconvex sets

This paper consists of two parts. In Section 2, we are mainly devoted to modify a combined homotopy interior point method (proposed by Yu and Lin, Homotopy method for a class of nonconvex Brouwer fixed point problems, Appl. Math. Comput. 74 (1996) 65–77) to solve fixed points of self-mapping in a broader class of nonconvex sets. Several numerical examples are presented to show the effectiveness of modifications. In Section 3, we modify their method to solve fixed points of self-mapping in the unbounded convex sets, a simple example is given to show its modification.

On the complexity of a combined homotopy interior method for convex programming

In [G.C. Feng, Z.H. Lin, B. Yu, Existence of an interior pathway to a Karush–Kuhn–Tucker point of a nonconvex programming problem, Nonlinear Anal. 32 (1998) 761–768; G.C. Feng, B. Yu, Combined homotopy interior point method for nonlinear programming problems, in: H. Fujita, M. Yamaguti (Eds.), Advances in Numerical Mathematics, Proceedings of the Second Japan–China Seminar on Numerical Mathematics, Lecture Notes in Numerical and Applied Analysis, vol. 14, Kinokuniya, Tokyo, 1995, pp. 9–16; Z.H. Lin, B. Yu, G.C. Feng, A combined homotopy interior point method for convex programming problem, Appl. Math. Comput. 84 (1997) 193–211.], a combined homotopy was constructed for solving non-convex programming and convex programming with weaker conditions, without assuming the logarithmic barrier function to be strictly convex and the solution set to be bounded. It was proven that a smooth interior path from an interior point of the feasible set to a K–K–T point of the problem exists. This shows that combined homotopy interior point methods can solve the problem that commonly used interior point methods cannot solve. However, so far, there is no result on its complexity, even for linear programming. The main difficulty is that the objective function is not monotonically decreasing on the combined homotopy path. In this paper, by taking a piecewise technique, under commonly used conditions, polynomiality of a combined homotopy interior point method is given for convex nonlinear programming.

A combined homotopy interior point method for the linear complementarity problem

A combined homotopy interior point method for the linear complementarity problem (LCP) is proposed in this paper. We construct a homotopy path for the linear complementarity problem and prove that the homotopy path determines a smooth interior path from an arbitrary given interior point to the solution of LCP. Furthermore, we modify a predictor corrector algorithm with the technique of the cone neighborhood to trace the homotopy path. Finally, a numerical example is provided to show the effectiveness and feasibility of this method.

A combined homotopy interior point method for convex nonlinear programming

In this paper, we present a new interior point method—combined homotopy interior point method (CHIP method) for convex nonlinear programming. Without strict convexity of the logarithmic barrier function and boundedness and nonemptiness of the solution set, we prove that for any ϵ > 0, an ϵ-solution of the problem can be obtained by the CHIP method. To our knowledge, strict convexity of the logarithmic barrier function and nonemptiness and boundedness of the solution set are the essential assumptions of the well-known center path-following method. Therefore, the CHIP method essentially reduces the assumptions of the center path-following method and can be applied to more general problems.

A combined homotopy interior point method for general nonlinear programming problems

A combined homotopy interior point method for solving general nonlinear programming is proposed. The algorithm generated by this method to Kuhn-Tucker points of the general nonlinear programming problems is proved to be globally convergent, under the “normal cone condition” about the constraints, probably without the convexity.