p-regular nonlinearity: tangency at singularity in degenerate optimization problems

We investigate description of the tangent cone to the null set of a mapping F at a given point x∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^{*}$$\end{document} in the case when F is degenerate at x∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^{*}$$\end{document}. To this aim we introduce the concept of modified 2-regular mappings, which generalizes the concept of p-regular mappings. Our main result provides the description of the tangent cone to the null set of modified 2-regular mappings. With the help of this result we derive new optimality conditions for a wide class of optimization problems with equality constraints.


Introduction
The problem of local description of the solution set appears in formulation of optimality conditions and construction of solution methods for optimization problems. In the present paper we consider degenerate optimization problems min ϕ(x) subject to F(x) = 0, B Ewa M. Bednarczuk where ϕ : X → R is defined on a Banach space X and the feasible solution set is described by a mapping F : X → Y , where Y is a Banach space, which is degenerate at the solution point x * , i.e. Im F (x * ) = Y .
Degenerate problems appear often in applications. It was shown in Marsden and Tretyakov (2003) that degeneracy (singularity) defined as at a given admissible point x * , F(x * ) = 0, is, in some sense, typical for nonlinear mappings F. Degeneracy occurs in the calculus of variations and the optimal control problems with boundary value conditions, e.g. in Chaplygin problem. The development of optimality conditions for degenerate problems is an active research topic, see Byrd et al. (1995), Dmitruk (1987), Schättler (1995, 1998a, b).
Here we focus on the description of the tangent cone where M(x * ) = {x ∈ X | F(x) = F(x * ) = 0} and ε > 0 is small enough. In the nondegenerate (regular) case, i.e., when ImF (x * ) = Y the problem of description of elements h ∈ X such that h ∈ T M(x * ) has been solved by the famous Lusternik's theorem which states that the tangent cone T M(x * ) to the set M(x * ) at the point x * coincides with the kernel of the derivative operator F (x * ), i.e., we have The degenerate case has been already investigated e.g., in Brezhneva and Tretyakov (2007), Buchner et al. (1983), Ledzewicz and Schättler (1998a), Ledzewicz and Schättler (1998b), Tretyakov (1984), where the constructive descriptions of the tangent cone to the null set M(x * ) are given for some classes of degenerate mappings. However, the classes of mappings considered so far, do not contain many important degenerate mappings. To enlarge the class of degenerate (singular) mappings with the constructive description of the tangent cone T M(x * ) to the set M(x * ) at the point x * we apply the tools of the p-regularity theory introduced and studied in Brezhneva and Tretyakov (2003), Marsden and Tretyakov (2003), Tretyakov (1984Tretyakov ( , 1983Tretyakov ( , 1987. The main idea of the p-regularity theory is to replace the operator F (x * ), which is not onto, with a linear operator p (x * , h), p ≥ 2, related to the p-th order Taylor's polynomial of F at x * , which is onto. The operator p (x * , h) contains the derivatives of F up to the p-th order, so in our considerations, F is assumed to be p-times continuously differentiable in a neighbourhood of x * . The order p is chosen as the smallest number for which the operator p is regular.
Let us point out that mathematical programing problems with complementarity constraints min ϕ(x) subject to are degenerate. Indeed, the constraints x i g i (x) = 0, i = 1, . . . , n are degenerate (nonregular) if x i and g i (x) are active at the solution point x * (the strict complementarity conditions do not hold). Then we can not apply the classical optimality conditions and moreover, Newton-type methods are inapplicable. It turns out, however, that the constraints of problem (1.2) are 2-regular along some element h ∈ X in the sense defined below, and thus, we are able to provide meaningful optimality conditions for (1.2) and to construct efficient Newton-type solution methods.
There are many papers devoted to the investigation of deformations and perturbations of optimization problems, see e.g. Jongen et al. ( , 1983, Jongen et al. (1990), Klatte and Kummer (2002), Rückmann (1993) and the references therein. Small perturbations of data of degenerate optimization problems may lead to large changes in solutions and/or to nonexistence of approximate solutions. It turns out that the existence of the p-regular structure of the problem under investigation entails stability of approximate solutions.
For these reasons, p-regularity theory is a valuable and adequate tool for providing optimality conditions and solution methods for large classes of degenerate optimization problems.

Elements of the p-regularity theory
Consider the equation where F : X → Y X, Y are Banach spaces, F ∈ C p+1 (X ). Let us assume that F (x * ) is degenerate (singular) at a given point x * ∈ M(x * ). In this section we recall basic constructions of p-regularity theory as developed in Brezhneva and Tretyakov (2003), Marsden and Tretyakov (2003), Tretyakov (1984), Tretyakov (1983), Tretyakov (1987) to investigate singular mappings. We assume that the space Y is decomposed into the direct sum where Y 1 := cl ImF (x * ), Z 1 = Y . Let Z 2 be a closed complementary subspace to Y 1 (we assume that such closed complement subspace exists), and let P Z 2 : Y → Z 2 be the projection operator onto Z 2 along Y 1 . By Y 2 we mean the closed linear span of the image of the map P Z 2 F (2) (x * )[·] 2 . More generally, we define inductively The order p is the smallest number for which (2.2) holds.
is called the p-factor operator of the mapping F at the point x * .

Definition 2.2
We say that the mapping F is p-regular at x * along (on) an element

Definition 2.3
We say that the mapping F is p-regular at x * if it is p-regular along any h ∈ X from the set

For the linear surjective operator
The following two theorems describe the tangent cone T M(x * ) to the set M(x * ) at the point x * and the null sets M(x * ) of p-regular and strongly p-regular mappings, respectively.
Theorem 2.5 (Brezhneva and Tretyakov 2007) Let X , Y be Banach spaces. Let F : Theorem 2.6 (Prusińska and Tretyakov 2016 The proof of Theorem 2.6 can be found in Prusińska and Tretyakov (2016). The importance of this result in the degenerate case is analogous to the importance of the classical implicit function theorem in nondegenerate case. In particular, Theorem 2.6 is used in proving optimality conditions for degenerate constrained optimization problems (Brezhneva and Tretyakov 2003) (see Sect. 4).

Generalization of p-regularity and description of the p-th order tangent cone
In Theorem 2.5, the crucial assumption which allows the constructive description of the tangent cone T M(x * ) to the set M(x * ) at x * is condition (2.4), i.e., the p-regularity of F along any element h ∈ H p (x * ). However, condition (2.4) may fail. Here In the sequel, we consider separately the cases where p = 2 and p ≥ 3.

Case p = 2
Suppose that the space Y is decomposed into the direct sum The order q is chosen as the smallest number for which condition (3.1) holds. Let us define the mappings is called the modified 2-factor-operator.

Definition 3.3 We say that the mapping
Example 3.4 (Continuation of Example 3.1) The following theorem gives the description of elements of the tangent cone T M(x * ) for modified 2-regular mappings.
Theorem 3.5 Let X, Y be Banach spaces. Let F : X → Y , F ∈ C 3 (X ) and F(x * ) = 0. Assume that F is modified 2-regular at x * along h 1 , . . . , h q and For the proof of Theorem 3.5 we need the Multivalued Contraction Mapping Principle (MCPP) proved in Brezhneva and Tretyakov (2007).

Then for any r
Moreover, among the pointsw satisfying (3.4) there exists a point such that Here H ( 1 , 2 ) is the Hausdorff distance between sets 1 and 2 , Proof of Theorem 3.5 For the sake of simplicity consider the case F (x * ) = 0. Set γ := 1 2 q and introduce the following operators Define h(t) := th 1 + · · · + t 1+(q−1)γ h q and consider the mapping We show that all assumptions of (MC M P) are satisfied for (x) with some ball We start by checking the assumption 2. of (MC M P). By the definition of (0), there exists c 1 ≥ 0 such that (3.5) By using Taylor's expansion we get for k = 1, . . . , q where ω k (t) ≤ ct 3 . By definition of mapping F 2,k , we have By (3.7) and (3.6), we obtain Then, by (3.5) and the latter relation with γ = 1 2q we obtain For sufficiently small t with some α ∈ (0, 1) and r 1 := r (t) = o(t 1+(q−1)γ ) we get (0) < (1 − α)r 1 which proves assumption 2. of (MC M P). Now we show that assumption 1. of (MC M P) holds for all x 1 , x 2 ∈ U r (t) (0) that is From this we deduce that . By using Mean Value Theorem and Taylor's expansion for k = 1, . . . , q, there exists c k > 0 such that Hence, with r 1 := r (t) = 0(t 1+γ (q−1) ) we get which proves assumption 1. of (MC M P). By (MC M P), there exists ω(t) such that ω(t) ∈ (ω(t)) which is equivalent to Hence, One can easily show that, by (3.5), and by the inequality ω(t) ≤ c (0) , we obtain the following estimate which finishes the proof.

Case p ≥ 3
We seek an element x(t) ∈ M(x * ) in the form For the sake of simplicity we assume that As previously, , Z 2 is closed complementary subspace to Y 1 and P Z 1 : Y → Z 1 , P Z 2 : Y → Z 2 are projection operators onto Z 1 and Z 2 , respectively, Y 2 := cl Im More generally, we define inductively

Definition 3.8 The linear operator
is called the modified p-factor operator.

Definition 3.10 The mapping F is modified p-regular at x
where α := 1 2 p and ω(t) = o(t 1+α ). The proof of the theorem below remains the same when α assumes any value from a given interval α ∈ (0, ε), ε < 1.

Theorem 3.11 Let X, Y be Banach spaces. Let
Assume that F is modified p-regular at x * along h 1 , h 2 and for the linear operator Then h 1 ∈ T M(x * ) and there exists ω(t), where c > 0 is an independent constant.
Proof The proof of this theorem is analogous to the proof of Theorem 3.5 and therefore we omit it.

Degenerate optimization problems
We consider the nonlinear optimization problem where ϕ : X → R is a sufficiently smooth function and F : X → Y is a sufficiently smooth mapping from a Banach space X into a Banach space Y . Let us consider the case where the mapping F is degenerate at the solution x * of problem (4.1) that is, when the derivative F (x * ) is not onto. In our previous works (Bednarczuk et al. 2011;Brezhneva and Tretyakov 2003) we derived optimality conditions for constrained optimization problems (4.1) that are p-regular at x * , i.e., when F is p-regular at x * . Now we use the results of the previous sections to prove optimality conditions for problems with mappings F which are strongly p-regular or modified 2-regular. Let us define p-factor Lagrange function The following optimality conditions for p-regular and strongly p-regular mappings F were proved in Brezhneva and Tretyakov (2003).
The proof of this theorem is based on Theorem 2.6 and can be found in Brezhneva and Tretyakov (2003). It turns out that there exist numerous problems for which the assumption of p-regularity of the mapping F fails at the solution x * .
Example 4.2 Consider the following problem We investigate optimality of x * := (0, 0, 0, 0) T . The mapping F is not 2-regular at x * and we cannot apply Theorem 4.1.
However, for modified 2-regular mappings F the following result holds. Let us introduce the modified 2-factor Lagrange function Theorem 4.3 (Case p = 2) Let X and Y be Banach spaces, F : X → Y , ϕ : X → R, ϕ ∈ C 2 (X ) and F ∈ C 3 (X ).
Assume that there exist elements h 1 , . . . , h q ∈ X such that the mapping F is modified 2-regular at x * along h 1 , . . . , h q and assumption 3.3 of Theorem 3.5 is fulfilled.
Then there exists a multiplier λ * ∈ Y * such that The proof of Theorem 4.3 is similar to the proof of Theorem 3.3 of Brezhneva and Tretyakov (2003) and we omit it here.
Based on the fact that for k = 1, . . . , p we obtain, analogously as in the proof of Theorem 3.5, that mapping p (x) satisfies all the assumptions of (MC M P) with some ball Br (t) (0), where r 1 := r (t) = o(t 1+α+ε ).

Conclusions
In this paper we derived new optimality conditions for problem with degenerate equality constraints. Our approach is based on constructions of p-regularity theory and on the modification of the concept of p-regularity. In Sect. 3 we proved a new theorem on the null set description and investigate the structure of the tangent cone for modified p-regular mappings. These results generalize the tangent cone descriptions obtained so far. Let us note that Theorems 3.5 and 3.11 do not give a complete description of the tangent cone T M(x * ) to the set M(x * ) at the point x * of the mapping F but knowing a single element h ∈ T M(x * ) is enough to prove optimality conditions for optimization problems (4.1) with the modified p-regular mappings F.
In Sect. 4 we derived new optimality conditions for modified p-regular constrained optimization problems. These results generalize necessary optimality conditions obtained for p-regular problems. The presented results can be considered as a part of the p-regularity theory.