A modified exact smooth penalty function for nonlinear constrained optimization

In this paper, a modified simple penalty function is proposed for a constrained nonlinear programming problem by augmenting the dimension of the program with a variable that controls the weight of the penalty terms. This penalty function enjoys improved smoothness. Under mild conditions, it can be proved to be exact in the sense that local minimizers of the original constrained problem are precisely the local minimizers of the associated penalty problem.

MSC: 47H20, 35K55, 90C30.

Merit function has always taken an important role in optimization problem. It is traditionally constructed to solve nonlinear programs by augmenting the objective function or a corresponding Lagrange function some penalty or barrier terms with respect of the constraints. Then it can be optimized by some unconstrained or bounded constrained optimization softwares or sequential quadratic programming (SQP) techniques. No matter what kind of techniques are involved, the merit function always depends on a small parameter ε or large parameter . As , the minimizer of a merit function such as a barrier function or the quadratic penalty function, converges to a minimizer of the original problem. By using some exact penalty function such as penalty function (see [1,9,20-22]), the minimizer of the corresponding penalty problem must be a minimizer of the original problem when ε is sufficiently small. There are some nonsmooth penalty functions for nonsmooth optimization problems, such as the exact penalty function using the distance function for the nonsmooth variational inequality problem in Hilbert spaces [18] and the one in [19].

The traditional exact penalty functions [8] are always nonsmooth. When it is used as a merit function to accept a new iterate in an SQP method, it may cause the Maratos effect [13]. On the other hand, a traditional smooth penalty function like the quadratic penalty function cannot be an exact one. So we must compute a sequence of minimization subproblem as . At that time, ill-conditioning may occur when the penalty parameter is too large or small, which also brings difficulty of computation. In [3] and [6], some kinds of augmented Lagrangian penalty functions have been proposed with improved exactness under strong conditions. In [11], exact penalty functions via regularized gap function for variational inequalities have also been given. All these functions enjoy some smoothness, but at the very beginning, to use this smoothness we need second-order or third-order derivative information of the problem function that is difficult to estimate in practice. Besides, all the above kinds of penalty functions (see [2-4,7,16] for summary) may be unbounded below even when the constrained problem is bounded, which may make it difficult to locate a minimizer.

In the paper [10], a new penalty function is proposed for the constrained optimization problem. By augmenting the dimension of the program with an additional variable ε that controls the weight of the penalty terms, this new penalty function enjoys properties of smoothness and exactness, and remains bounded below under reasonable conditions. Its important new idea is that the penalty function is considered as a function of variable x and the additional variable ε simultaneously. Under proper assumptions, the minimizer of the merit function satisfies , and is a minimizer of the original problem. However, the penalty function given in [10] is not smooth in a small neighborhood of , where the minimizer of the original constrained problem lies. In this paper, we give a penalty function which enjoys the properties of the penalty function given in [10] and has improved smoothness.

The rest of this paper is organized as follows. In Section 2, a penalty function is introduced for a smooth nonlinear optimization problem with equality constraints and bounded constraints. The smoothness of this penalty function is discussed, as well as other properties, including being bounded below under mild assumptions. Section 3 shows the exactness of our penalty function in the sense that under certain conditions, local minimizer of our penalty function has the form with and is a local minimizer of the original problem, and a converse result holds.

Notation Throughout this paper, we use the Euclidean norm . The subvector of x indexed by the indices in J is denoted by . We denote sets of the form

where the lower bound and the upper bound are vectors containing proper or infinite bounds on the components of x and is referred to an n-dimensional box.

We consider the smooth nonlinear optimization problem with equality constraints and bound constraints:

where is a box in with nonempty interior, and are continuously differentiable in an open set D containing and . We fix and consider the equivalent problem:

where .

Let be an upper bound of the parameter ε. Then the corresponding penalty function on for () is given as follows:

with the constraint violation measure

where, in addition, and , are all fixed numbers, is a penalty parameter and is a Euclidean norm, with for any vector x.

Obviously, if and only if , , . The corresponding penalty problem then reads

The main difference between (2.3) and the penalty function given in [10] is that in (2.3), , which does not have the property that , as .

It is easy to see that is continuously differentiable with respect to on

2.1 Boundedness of the penalty function

If , , then

Therefore, is bounded below on the set

whenever is bounded below on the set . This is a reasonable condition since it usually holds when f is bounded below on the feasible set, is small enough, and q is large enough.

The denominator is included since it forces the level sets of to remain in the set , hence in some sense does not go far away from the feasible set of (P).

Now we see a simple example:

It has a bounded feasible domain, a global minimizer at with , and a local minimizer . The traditional quadratic penalty function for this problem

is unbounded below for all penalty parameters since, e.g., for , . It is also the case for traditional penalty functions, including multiplier penalty functions that use an additional term . On the other hand, our new penalty function is bounded below. Set , it reads

where . Since , if , it is obvious that our penalty function is bounded below. See Figure 1 for the display of the contour of the penalty function on this example.

In this section, we show that our penalty function is exact in the sense that under certain conditions, local minimizer of our penalty function has the form in which and is a local minimizer of the original problem and a converse proposition holds.

Firstly, recall the Mangasarian-Fromovitz condition. We say that the Mangasarian-Fromovitz condition (see[12]) for Problem (P) holds atifhas full rank and there is a vectorwithand

Theorem 3.1Assume that the setdefined in Section 2 is bounded, and the Mangasarian-Fromovitz condition holds at each. Letin (). Ifσis sufficiently large, then there is no Kuhn-Tucker pointof () with.

Proof Let the Lagrangian function of () for be

where , are the Lagrangian multipliers. If is a Kuhn-Tucker point of () with , then there exist vectors such that

then

and

where is the gradient of with respect to . The assertion of the theorem is proved by contradiction. □

Assume that there exists a sequence with for all k, as , where is a Kuhn-Tucker point of (). We use the abbreviation . The point satisfies

hence, . Since is closed and bounded, we may restrict ourselves to a subsequence if necessary and assume that

The condition yields

with equality in the case . When and because - the right side of (3.2) is a finite number, we have

where . On the other side, the derivative the penalty function with respect to x is given as

(3.4) is equivalent to

Let , since or , we have

Because , thus Mangasarian-Fromovitz condition holds at and there exists some vector such that , where p satisfies (3.1). Let , and . Then by (3.6), we have

(3.7) and the Mangasarian-Fromovitz condition (3.1) imply for . Thus, . Now the fact that has full rank yields , i.e.,

and by , it follows that

By (3.3), we obtain

Thus, .

Furthermore, by (3.2), it holds that

Let , the last term on the left-hand side tend to +∞. Thus, the vectors satisfies . The vectors have norm 1, and (3.5) implies that the numbers (), defined by

satisfy

If we pick a convergent subsequence with the limit and pass to the limit we obtain

Now similarly as above, it yields , which is a contradiction with . Thus such a sequence cannot exist, and for sufficiently large , all Kuhn-Tucker points of () are of the form .

Theorem 3.2Assume thatis a local minimizer of minimizer of () with finite, whereis sufficiently large. If the hypotheses of Theorem 3.1 are fulfilled, thenis a local minimizer of (P).

Proof Now let be a local minimizer of () with finite and is sufficiently large. If , then must be a Kuhn-Tucker point of (), which is a contradiction with Theorem 3.1. Therefore, , and since is finite, . It implies that , and by (2.5) there is a neighborhood of where for feasible x. Therefore, is a local minimizer of (P). □

We now show a converse result of Theorem 3.2, which will use the following lemmas.

Lemma 3.1Suppose, andhas full rank. Then there exist a neighborhoodofand a constantsuch that for each, and each subsetJof, there exists a vectorwith, forand, , such that

Proof Since and has full rank, there exists a matrix such that the augmented matrix is nonsingular. By the continuity of at , there exists a neighborhood of such that is nonsingular, for any . Take for the closed convex hull of , then for all , the matrix is nonsingular. We now show that for any , there exists a matrix such that

In fact, given , it follows from the mean value theorem that

where , so (3.8) holds. Set the mapping , for . By the proof in [10], Theorem 4.5], we have that there exists a neighborhood of such that for each , and each subset J of , there exists a vector with

so , for and , .

For , we have

for some . On the other side, we have

Therefore, combining (3.8) with (3.9), we have

where

and this complete the proof. □

Lemma 3.2Assume thatis a local minimizer of problem (2.1) with, , where. Suppose thathas full rank. Then there exists a constantsuch that

whereis defined in Lemma 3.1.

Proof From Lemma 3.1, let and be the one in Lemma 3.1. Let , then by Lemma 3.1 with , there exists an with and , such that

So y is a feasible point of problem (2.1), and . Since f is continuously differentiable, for any , there exists a vector such that

where ξ lies in the segment between x and y. Set , we have

where the last inequality holds from (3.10).

Let , then

which complete the proof. □

Theorem 3.3Ifis a local minimizer of problem (P) with, , where, andhas full rank, then for sufficiently large, there are a neighborhoodofand asuch that

In particular, is a local minimizer of.

Proof Let is a neighborhood of such that

where is defined in Lemma 3.1.

For , where and , we distinguish two cases.

Case 1. . For this case, we have

where the second inequality is by (3.11).

Case 2. . Then , and

The last inequality holds since .

Let , we get

From Case 1 and Case 2, we obtain the conclusion. □

In this paper, a modified exact penalty function for equality constrained nonlinear programming problem is constructed by augmenting a new variable that controls the constraint violence. This function enjoys smoothness, and with very mild conditions it is proved to be an exact penalty function.

Since in practice, a lot of applied problems are nonsmooth, it is a meaningful work to extend the results in this paper to the nonsmooth case. By using the limiting subgradients that is presented in two books written by Mordukhovich [14,15], as well as Clarke’s generalized gradients in [5], we can extend the penalty function with the mentioned good properties to nonsmooth optimization problems, just as that has been done in [17-19]. That will be our future research direction.

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

The authors wish to thank the anonymous referees for their endeavors and valuable comments. The authors would also like to thank Professor Zhang Liansheng for some very helpful comments on a preliminary version of this paper. This research was supported by the National Natural Science Foundation of China under Grants 10971118, 11101248, Natural Science Foundation of Shandong Province under Grants ZR2012AM016, and the foundation 4041-409012 of Shandong University of Technology.

1. Antczak, T: Exact penalty functions method for mathematical programming problems involving invex functions. Eur. J. Oper. Res.. 198, 29–36 (2009). Publisher Full Text

2. Bazaraa, MS, Sherali, HD, Shetty, CM: Nonlinear Optimization Theory and Algorithms, Wiley, New York (1993)

3. Bertsekas, DP: Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York (1982)

4. Boukari, D, Fiacco, AV: Survey of penalty, exact-penalty and multiplier methods from 1968 to 1993. Optimization. 32, 301–334 (1995). Publisher Full Text

5. Clarke, FH: Optimization and Nonsmooth Analysis, Wiley-Interscience, New York (1983)

6. Di Pillo, G: Exact penalty methods. In: Spedicato E (ed.) Algorithms for Continuous Optimization, pp. 209–253. Kluwer Academic, Dordrecht (1994)

7. Fletcher, R: Practical Methods of Optimization, Wiley, New York (1987)

8. Han, SP, Mangasarian, OL: Exact penalty functions in nonlinear programming. Math. Program.. 17, 251–269 (1979). Publisher Full Text

9. Hoheisel, T, Kanzow, C, Outrata, J: Exact penalty results for mathematical programs with vanishing constraints. Nonlinear Anal.. 72, 2514–2526 (2010). Publisher Full Text

10. Huyer, W, Neumaier, A: A new exact penalty function. SIAM J. Optim.. 13, 1141–1158 (2003). Publisher Full Text

11. Li, W, Peng, J: Exact penalty functions for constrained minimization problems via regularized gap function for variational inequality. J. Glob. Optim.. 37, 85–94 (2007)

12. Mangasarian, OL, Fromovitz, S: The Fritz John necessary optimality conditions in the presence of equality and inequality constraints. J. Math. Anal. Appl.. 17, 37–47 (1967). Publisher Full Text

13. Maratos, N: Exact penalty function algorithms for finite dimensional and control optimization problems. PhD thesis, University of London (1978)

14. Mordukhovich, BS: Variations Analysis and Generalized Differentiation, I: Basic Theory, Springer, Berlin (2006)

15. Mordukhovich, BS: Variations Analysis and Generalized Differentiation, II: Applications, Springer, Berlin (2006)

16. Nocedal, J, Wright, SJ: Numerical Optimization, Springer, New York (1999)

17. Soleimani-damaneh, M: The gap function for optimization problems in Banach spaces. Nonlinear Anal.. 69, 716–723 (2008). Publisher Full Text

18. Soleimani-damaneh, M: Penalization for variational inequalities. Appl. Math. Lett.. 22, 347–350 (2009). Publisher Full Text

19. Soleimani-damaneh, M: Nonsmooth optimization using Mordukhovich’s subdifferential. SIAM J. Control Optim.. 48, 3403–3432 (2010). Publisher Full Text

20. Zangwill, WI: Nonlinear programming via penalty function. Manag. Sci.. 13, 344–358 (1967). Publisher Full Text

21. Zaslavski, AJ: A sufficient condition for exact penalty in constrained optimization. SIAM J. Optim.. 16, 250–262 (2005). Publisher Full Text

22. Zaslavski, AJ: Stability of exact penalty for nonconvex inequality-constrained minimization problems. Taiwan. J. Math.. 14, 1–19 (2010)