Solving nonlinear programming problems with unbounded non-convex constraint sets via a globally convergent algorithm
Journal of Inequalities and Applications volume 2014, Article number: 281 (2014)
In this paper, we propose a set of unbounded conditions, under which we are able to solve nonlinear programming problems in a class of unbounded non-convex sets via the combined homotopy interior point algorithm. We also obtain the global convergence of the combined homotopy interior point algorithm and analyze the implementation of this algorithm in detail.
Consider the constrained minimization problem:
where , , and , are three times continuously differentiable.
We call a point is a Karush-Kuhn-Tucker (K-K-T) point of (1) and , are the corresponding Lagrangian multiplier vectors if satisfies
where , , , , , , and .
Since Kellogg et al. (see ) and Smale (see ) proposed the notable homotopy method, this method has become a powerful tool in dealing with various nonlinear problems, for example, zeros or fixed points of maps (see [3–5], etc. and the references therein). However, the homotopy method has seldom been touched upon in constrained optimization until 1988, Megiddo (see ) and Kojima et al. (see ) discovered the Karmarkar interior point method for linear programming was a kind of path-following method. From then on, the central path-following methods for mathematical programming have become an active research subject. Furthermore, it was extended to convex nonlinear programming problems recently (see [8–11], etc.). But all their convergence results were obtained under the assumptions that the logarithmic barrier function is strictly convex and the solution set is nonempty and bounded.
Recently, a combined homotopy interior point method (denoted as CHIP method for convenience) for nonlinear programming problems was presented in [12, 13] (detailed abstract of them was announced in ). In , compared with the central path-following methods, the authors removed the convexity of the logarithmic barrier function and the boundedness and nonemptiness of the solution set. This shows that the CHIP method can solve the problems that interior path-following methods cannot solve. In , by taking a piecewise technique, under the commonly used conditions, the polynomiality of the CHIP method was given, which shows that the efficiency of the CHIP method is also very well. In , we introduced functions , and , to extend the results in [12, 17] to more general non-convex sets. However, there are no results reported in the literature about the work in  extended to unbounded non-convex sets. In this paper, we attempt to complete this work. To this end, by using some inequality techniques and the ideas of infinite solutions which were introduced in [18, 19], we develop a set of new unboundedness conditions. Under these conditions, we obtain the global convergence results of the CHIP method and therefore extend the work in  to unbounded non-convex sets.
2 Main results
In this section, the nonnegative and positive orthants of are denoted as and , respectively. We also denote by the active set at x. In addition, let be the feasible set, let be the strictly feasible set and let be the boundary set of X.
In , to solve (2), the following homotopy was constructed:
where , , and .
In this section, we utilize the concept of infinite solutions [18, 19] and hence give a new set of unboundedness conditions. The nonlinear programming problem is said to have a solution at infinite, if there exists a sequence satisfying , as , and for any given , there exist and such that
Then we assume that there exist smooth mappings , and , such that:
(A1) is nonempty; nonlinear programming problems have no infinite solutions.
(A2) , if
then , and , .
(A3) , we have
(A4) , is of full of column rank and is nonsingular.
However, only applying the infinite solution technique to the items , , and , we cannot extend the results in  to unbounded cases because of the existence of the items and . So for the items and , we need to use other techniques. In this paper, we still need the following assumption:
For a given , rewrite as . The zero-point set of is
The inverse image theorem tells us that, if 0 is a regular value of the map , then consists of some smooth curves. The regularity of can be obtained by the following lemma.
Lemma 2.1 (Parameterized Sard theorem)
Let , be open sets, and a map, where . If is a regular value of Φ, then for almost all , 0 is a regular value of .
Lemma 2.2 Let H be defined as in (3). In addition, let assumptions (A1)-(A5) hold, let , and , be functions, and let , and , be functions. Then, for almost all , 0 is a regular value of , and consists of some smooth curves, among which there exists a smooth curve that starts from .
Lemma 2.3 Let H be defined as in (3). In addition, let assumptions (A1)-(A5) hold, let , and , be functions, and let , and , be functions. Then, for almost all , the projection of the smooth curve onto the x-plane is bounded.
Proof If not, then there exists a sequence of points such that as .
It is easy to show that the following inequality holds:
By the homotopy equation (3), we have
Multiplying the first equation in (5) by , we get
Since and , then . Besides, by the third equation in (5), we have , thus the following inequality holds:
Then by (9), we have
When , by (10), we have
which contradicts assumption (A1). □
Theorem 2.1 Let H be defined as in (3), let , , and , be three times continuously differentiable functions, let assumptions (A1)-(A5) hold, and let , and , be twice times continuously differentiable functions. Then for almost all , there exists a curve of dimension 1 such that
When , tends to a point . In particular, the component of is a K-K-T point of problem (1).
Proof By Lemma 2.2, there must be a curve of dimension 1 (denoted by ) such that
By the classification theorem of one-dimensional smooth manifolds, is diffeomorphic either to a unit circle or to a unit interval. For any , is nonsingular, so cannot be diffeomorphic to a unit circle. That is, is diffeomorphic to a unit interval.
Let be a limit point of . If , because 0 is a regular value of , , and the Jacobian matrix of H at is of full row rank, then by the implicit function theorem, can be extended at . This result contradicts the fact that is a limit point of .
Let . Thus, and the following three cases are possible:
Since the equation has a unique solution in , case (b) is impossible.
By the homotopy equation (3), we have
If , then
Because X and are bounded, by assumption (A2), the third part on the left-hand side of (16) tends to infinity as , but the other two parts are bounded, this is impossible. Hence we conclude that the projection of the smooth curve onto the z-plane is also bounded.
In case (c), first, we prove that . If , then there exist and a sequence of points such that as . From (15), we have
Because X and are bounded, when , the left-hand side of (17) tends to 0. At the same time, the right-hand side of (17) tends to , which is strictly less than 0. This fact results in a contradiction.
Then we only need to prove that the remainder of case (c) is impossible. If not, then there exists a sequence of points such that for some as . From (15), we obtain . Because X and are bounded, there exists a subsequence of points (denoted also by ) such that , , , and as .
When , from (15), the active index set is . When , the index set .
(1) When , from (16), by the fact that is bounded for each , assumptions (A1)-(A2), we conclude that and . Therefore, when , (16) becomes
which contradicts assumption (A3).
(2) When , rewrite (16) as
Let be a vector with components , . Then set
Note that ; then there exists a subsequence of , still denoted by , such that for each as . Furthermore, the vector with components , is denoted by ; thus, . Dividing both sides of (19) by and letting , we have
which contradicts assumption (A2).
(3) When , because the nonempty index set , the proof of (3) is similar to that of (2).
From the discussion above, we conclude that (a) is the only possible case. Hence, the x-component of is a K-K-T point of (1). □
3 Algorithmic analysis
For almost all , by Theorem 2.1, the homotopy generates a curve , by differentiating the first equation in (12), we obtain the following theorem.
Theorem 3.1 The homotopy path is determined by the following initial value problem to the ordinary differential equation:
where s is the arc length of the curve .
As for how to trace numerically the homotopy path, there have been many predictor-corrector algorithms, see , etc. for reference. Hence we omit them in this paper. In the implementation of the algorithm, generally we need to be devoted to finding the positive direction of the tangent vector at a point on which keeps the sign of the determinant invariant. On the first iteration, the sign is determined by the following lemma.
Lemma 3.1 If is smooth, then the positive direction at the initial point satisfies .
where , . The tangent vector of at satisfies
where , .
It is easy to get , then
Since , , and , so the sign of
is . □
The following pseudocode describes the basic steps of a generic predictor-corrector method.
Algorithm 3.1 (Euler-Newton method)
Step 0. Provide an initial guess , an initial step length , and three small positive numbers , , and . Set .
Step 1. Compute the direction of the predictor step.
Compute a unit tangent vector .
Determine the direction of the predictor step as follows:
If the sign of the determinant is , then .
If the sign of the determinant is , then .
Step 2. Compute a corrector point .
If , then let , and go to Step 3.
If , then let , and go to Step 3.
If , then let , , and go to Step 2.
Step 3. If , then stop. Otherwise, , and go to Step 1.
In this paper, we present a set of unboundedness conditions, under which, we are able to solve nonlinear programming problems on a class of unbounded non-convex sets via the combined homotopy interior point algorithm. The main advantage of the algorithm presented in this paper is that it is a globally convergent algorithm whose initial points can be chosen more easily than the locally convergent algorithms. Since nonlinear programming problems have wide applications in engineering, management, economics, and so on, our results may be useful to propose a powerful solution tool for dealing with these nonlinear problems. In future, we devote our efforts to proposing new techniques to solve nonlinear programming problems in a broader class of unbounded non-convex sets.
Kellogg RB, Li TY, Yorke JA: A constructive proof of the Brouwer fixed-point theorem and computational results. SIAM J. Numer. Anal. 1976, 13: 473–483. 10.1137/0713041
Smale S: A convergent process of price adjustment and global Newton method. J. Math. Econ. 1976, 3: 1–14. 10.1016/0304-4068(76)90002-1
Chow SN, Mallet-Paret J, Yorke JA: Finding zeros of maps: homotopy methods that are constructive with probability one. Math. Comput. 1978, 32: 887–899. 10.1090/S0025-5718-1978-0492046-9
Garcia CB, Zangwill WI: An approach to homotopy and degree theory. Math. Oper. Res. 1979, 4: 390–405. 10.1287/moor.4.4.390
Li Y, Lin ZH: A constructive proof of the Poincaré-Birkhoff theorem. Trans. Am. Math. Soc. 1995, 347: 2111–2126.
Megiddo N: Pathways to the optimal set in linear programming. In Progress in Mathematical Programming, Interior Point and Related Methods. Edited by: Megiddo N. Springer, New York; 1988:131–158.
Kojima M, Mizuno S, Yoshise A: A primal-dual interior point algorithm for linear programming. In Interior Point and Related Methods. Edited by: Megiddo N. Springer, New York; 1988:29–47.
Kortanek KO, Potra F, Ye Y: On some efficient interior point algorithms for nonlinear convex programming. Linear Algebra Appl. 1991, 152: 169–189.
McCormick GP: The projective SUMT method for convex programming. Math. Oper. Res. 1989, 14: 203–223. 10.1287/moor.14.2.203
Monteiro RDC, Adler I: An extension of Karmarkar type algorithm to a class of convex separable programming problems with global linear rate of convergence. Math. Oper. Res. 1990, 15: 408–422. 10.1287/moor.15.3.408
Zhu JA: A path following algorithm for a class of convex programming problems. ZOR-Methods Models Oper. Res. 1992, 36: 359–377. 10.1007/BF01416235
Feng GC, Lin ZH, Yu B: Existence of interior pathway to the Karush-Kuhn-Tucker point of a nonconvex programming problems. Nonlinear Anal. 1998,32(6):761–768. 10.1016/S0362-546X(97)00516-6
Lin ZH, Yu B, Feng GC: A combined homotopy interior point method for convex nonlinear programming. Appl. Math. Comput. 1997, 84: 193–211. 10.1016/S0096-3003(96)00086-0
Feng GC, Yu B: Combined homotopy interior point method for nonlinear programming problems. Lecture Notes in Numerical and Applied Analysis 14. In Advances in Numerical Mathematics: Proceedings of the Second Japan-China Seminar on Numerical Mathematics. Edited by: Fujita H, Yamaguti M. Kinokuniya, Tokyo; 1995:9–16.
Yu B, Xu Q, Feng GC: On the complexity of a combined homotopy interior method for convex programming. J. Comput. Appl. Math. 2007,200(1):32–46. 10.1016/j.cam.2005.12.018
Su ML, Lv XR: Solving a class of nonlinear programming problems via a homotopy continuation method. Northeast. Math. J. 2008,24(3):265–274.
Liu QH, Yu B, Feng GC: A combined homotopy interior point method for nonconvex nonlinear programming problems under quasi normal cone conditions. Acta Math. Appl. Sin. 2003, 26: 372–377.
Xu Q, Yu B, Feng GC, Dang CY: Condition for global convergence of a homotopy method for variational inequality problems on unbounded sets. Optim. Methods Softw. 2007, 22: 587–599. 10.1080/10556780600887883
Xu Q, Lin ZH: The combined homotopy convergence in unbounded set. Acta Math. Appl. Sin. 2004, 27: 624–631.
Allgower EL, Georg K: Introduction to Numerical Continuation Methods. SIAM, Philadelphia; 2003.
This work was supported by NSFC-Union Science Foundation of Henan (No. U1304103) and Natural Science Foundation of Henan Province (No. 122300410261).
The authors declare that they have no competing interests.
MS carried out the main work and drafted the manuscript. JL participated in completing the proof of Lemma 2.3. ZX participated in completing the proof of Theorem 2.1. All authors read and approved the final manuscript.
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Su, M., Li, J. & Xu, Z. Solving nonlinear programming problems with unbounded non-convex constraint sets via a globally convergent algorithm. J Inequal Appl 2014, 281 (2014). https://doi.org/10.1186/1029-242X-2014-281
- unbounded conditions
- nonlinear programming problems
- non-convex sets