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

Abstract

We investigate description of the tangent cone to the null set of a mapping F at a given point $x^{*}$ in the case when F is degenerate at $x^{*}$. 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.

1 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

$$\begin{aligned} \min \ \varphi (x)\ \text {subject to}\,F(x)=0, \end{aligned}$$

where $\varphi :X\rightarrow \mathbb {R}$ is defined on a Banach space X and the feasible solution set is described by a mapping $F:X\rightarrow Y$, where Y is a Banach space, which is degenerate at the solution point $x^{*}$, i.e. $\text { Im }F'(x^{*})\ne Y$.

Degenerate problems appear often in applications. It was shown in Marsden and Tretyakov (2003) that degeneracy (singularity) defined as

$$\begin{aligned} \text {Im} F'(x^{*})\ne Y \end{aligned}$$

(1.1)

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), Ledzewicz and Schättler (1995, 1998a, b).

Here we focus on the description of the tangent cone

$$\begin{aligned} TM(x^{*})=\{h\in X\ |\ x^{*}+\alpha h+r(\alpha )\in M(x^{*}),\ \alpha \in [0,\varepsilon ],\ \Vert r(\alpha \Vert =o(\alpha )\}, \end{aligned}$$

where $M(x^{*})=\{x\in X\ |\ F(x)=F(x^{*})=0\}$ and $\varepsilon >0$ is small enough. In the nondegenerate (regular) case, i.e., when

$$\begin{aligned} \text {Im} F'(x^{*})= Y \end{aligned}$$

the problem of description of elements $h\in X$ such that $h\in TM(x^{*})$ has been solved by the famous Lusternik’s theorem which states that the tangent cone $TM(x^{*})$ to the set $M(x^{*})$ at the point $x^{*}$ coincides with the kernel of the derivative operator $F'(x^{*})$, i.e., we have

$$\begin{aligned} TM(x^{*})=\text {Ker} F'(x^{*}). \end{aligned}$$

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 $TM(x^{*})$ 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 (1984, 1983, 1987).

The main idea of the p-regularity theory is to replace the operator $F'(x^{*})$, which is not onto, with a linear operator $\Psi _{p}(x^{*},h)$, $p\ge 2$, related to the p-th order Taylor’s polynomial of F at $x^{*}$, which is onto. The operator $\Psi _{p}(x^{*},h)$ 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 $\Psi _{p}$ is regular.

Let us point out that mathematical programing problems with complementarity constraints

$$\begin{aligned} \begin{array}{l} \min \ \varphi (x)\ \text {subject to}\\ g_{1}(x)\ge 0,\ldots , g_{n}(x)\ge 0\\ x_{1}\ge 0\,\ldots , x_{n}\ge 0\\ x_{i}g_{i}(x)=0,\ i=1,\ldots ,n \end{array} \end{aligned}$$

(1.2)

are degenerate. Indeed, the constraints $x_{i}g_{i}(x)=0$, $i=1,\ldots ,n$ 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 $h\in X$ 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. (1986, 1983, 1986), 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.

2 Elements of the p-regularity theory

Consider the equation

$$\begin{aligned} F(x)=0, \end{aligned}$$

(2.1)

where $F:X\rightarrow Y$ X, Y are Banach spaces, $F\in C^{p+1}(X)$. Let us assume that $F'(x^{*})$ is degenerate (singular) at a given point $x^{*}\in M(x^{*})$. 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

$$\begin{aligned} Y=Y_{1}\oplus Y_{2}\oplus \cdots \oplus Y_{p}, \end{aligned}$$

(2.2)

where $Y_{1}:=\text{ cl } \text{ Im } F'(x^{*})$, $Z_{1}=Y$. Let $Z_{2}$ be a closed complementary subspace to $Y_{1}$ (we assume that such closed complement subspace exists), and let $P_{Z_{2}}:Y\rightarrow Z_{2}$ 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 $P_{Z_{2}}F^{(2)}(x^{*})[\cdot ]^{2}$. More generally, we define inductively $Y_{i}:=\text{ cl } (\text {Span Im}P_{Z_{i}}F^{(i)}(x^{*})[\cdot ]^{i})\subset Z_{i}$, $i=2,\ldots ,p-1$, where $Z_{i}$ is a chosen closed complementary subspace for $(Y_{1}\oplus Y_{2}\cdots \oplus Y_{i-1})$ with respect to Y, $i=2,\ldots ,p$ and $P_{Z_{i}}:Y\rightarrow Z_{i}$ is the projection operator onto $Z_{i}$ along $(Y_{1}\oplus Y_{2}\cdots \oplus Y_{i-1})$ with respect to Y, $i=2,\ldots ,p$. Finally, $Y_{p}=Z_{p}$. The order p is the smallest number for which (2.2) holds.

Let $F_{i}:X\rightarrow Y_{i}$, $i=1,\ldots ,p$ be defined as $F_{i}(x):=P_{i}F(x),$ where $P_{i}:= P_{Y_{i}}: Y\rightarrow Y_{i}$ is the projection operator onto $Y_{i}$ along $(Y_{1}\oplus Y_{2}\cdots \oplus Y_{i-1}\oplus Y_{i+1}\oplus ,\cdots \oplus Y_{p})$ with respect to Y, $i=1,\ldots ,p$.

Definition 2.1

The linear operator $\Psi _{p}(x^{*},h)\in \mathcal{L}(X, Y_{1}\oplus Y_{2}\oplus \cdots \oplus Y_{p})$ for $h\in X$, $h \ne 0$,

$$\begin{aligned} \Psi _{p}(x^{*},h):= F'_{1}(x^{*})+ F''_{2}(x^{*})[h]+\cdots + F^{(p)}_{p}(x^{*})[h]^{p-1} \end{aligned}$$

(2.3)

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 $h\in X$ if

$$\begin{aligned} \text {Im} \Psi _{p}(x^{*},h)=Y. \end{aligned}$$

(2.4)

Definition 2.3

We say that the mapping F is p-regular at $x^{*}$ if it is p-regular along any $h\in X$ from the set

$$\begin{aligned} H_{p}(x^{*}):=\bigcap _{k=1}^{p}\text {Ker}^{k}F_{k}^{(k)}(x^{*})\setminus \{0\}, \end{aligned}$$

where $\text {Ker}^{k} F_{k}^{(k)}(x^{*})=\{\xi \in X\ |\ F_{k}^{(k)}(x^{*})[\xi ]^{k}=0\}$ is the k-kernel of the k-order mapping $F_{k}^{(k)}(x^{*})[\xi ]^{k}$.

For the linear surjective operator $\Psi _{p}(x^{*},h):X\rightarrow Y$, by $\{\Psi _{p}(x^{*},h)\}^{-1}$ we denote its right inverse, $\{\Psi _{p}(x^{*},h)\}^{-1}:Y\rightarrow 2^{X}$, and we have

$$\begin{aligned} \{\Psi _{p}(x^{*},h)\}^{-1}y=\{x\in X\ |\ \Psi _{p}(x^{*},h)x=y\}. \end{aligned}$$

We define the norm of $\{\Psi _{p}(x^{*},h)\}^{-1}$ by the formula

$$\begin{aligned} \Vert \{\Psi _{p}(x^{*},h)\}^{-1}\Vert =\sup _{\Vert y\Vert =1}\inf \{\Vert x\Vert \ |\ x\in \{\Psi _{p}(x^{*},h)\}^{-1}y\}. \end{aligned}$$

We say that $\{\Psi _{p}(x^{*},h)\}^{-1}$ is bounded if $\Vert \{\Psi _{p}(x^{*},h)\}^{-1}\Vert <+\infty$.

Definition 2.4

The mapping F is called strongly p-regular at the point $x^{*}$ if there exists $\gamma >0$ such that

$$\begin{aligned} \sup _{h\in H_{\gamma }}\Vert \{\Psi _{p}(x^{*},h)\}^{-1}\Vert <+\infty , \end{aligned}$$

where

$$\begin{aligned} H_{\gamma }=\{h\in X\ |\ \Vert F_{k}^{(k)}(x^{*})[h]^{k}\Vert _{Y_{k}}\le \gamma ,\ k=1,\ldots ,p,\ \Vert h\Vert =1\}. \end{aligned}$$

The following two theorems describe the tangent cone $TM(x^{*})$ 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 $F:X\rightarrow Y$, $F\in C^{p+1}(X)$, $F(x^{*})=0$ and let F be p-regular at $x^{*}$. Then

$$\begin{aligned} TM(x^{*})=H_{p}(x^{*}). \end{aligned}$$

Theorem 2.6

(Prusińska and Tretyakov 2016) Let X, Y be Banach spaces. Let $F:X\rightarrow Y$, $F\in C^{p+1}(X),$ $p\ge 2$, $F(x^{*})=0$, and let F be strongly p-regular at $x^{*}$.

Then there exists a neighbourhood $U(x^{*})$, a mapping $\xi \rightarrow x(\xi ):U(x^{*})\rightarrow X$ and a constant $\delta >0$ such that

$$\begin{aligned} \begin{array}{l} F(\xi +x(\xi ))=0\\ \Vert x(\xi )\Vert \le \delta \sum _{k=1}^{p}\frac{\Vert F_{k}(\xi )\Vert _{Y_{k}}}{\Vert \xi -x^{*}\Vert ^{k-1}}\\ \text {and}\ \Vert x(\xi )\Vert \le \delta \sum _{k=1}^{p}\Vert F_{k}(\xi )\Vert _{Y_{k}}^{1/k} \end{array} \end{aligned}$$

for all $\xi \in U(x^{*})$.

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).

3 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 $TM(x^{*})$ to the set $M(x^{*})$ at $x^{*}$ is condition (2.4), i.e., the p-regularity of F along any element $h\in H_{p}(x^{*})$. However, condition (2.4) may fail. For $p=2$, there are examples such that $F''(x^{*})h\cdot X\ne Y$ ($F'(x^{*})=0$), where $h\in \text {Ker}^{2}F''(x^{*})$.

Example 3.1

Consider the mapping $F:\mathbb {R}^{4}\rightarrow \mathbb {R}^{2}$,

$$\begin{aligned} F(x):=\left( \begin{array}{l} x_{1}x_{2}-x_{3}^{2}\\ x_{3}x_{4}\end{array}\right) ,\ h=(1,0,0,0)^{T}\in \text {Ker}^{2}F''(0). \end{aligned}$$

At $x^{*}=0$, the 2-factor-operator $F''(0)h=\left( \begin{array}{llll} 0&{}1&{}0&{}0\\ 0&{}0&{}0&{}0\end{array}\right)$ is singular. Therefore, Theorem 2.5 does not apply. However, $h\in TM(0)$, i.e. $th+\omega (t)\in M(0)$ $\Leftrightarrow$ $F(th+\omega (t))=0$, $\Vert \omega (t)\Vert =o(t)$, $t\in [0,\varepsilon ]$, $\varepsilon >0$ sufficiently small. Here $\omega (t)=(0,0,0,0)^{T}$.

In the sequel, we consider separately the cases where $p=2$ and $p\ge 3$.

3.1 Case $p=2$

Suppose that the space Y is decomposed into the direct sum

$$\begin{aligned} Y=Y_{1}\oplus Y_{1}^{2}\oplus \cdots \oplus Y^{2}_{q}, \end{aligned}$$

(3.1)

where $Y_{1}=\text {cl Im } F'(x^{*})$, $Z_{1}=Y$, $Z_{1}^{2}$ is a closed complementarity subspace to $Y_{1}$ and let $P_{1}^{2}:=P_{Z_{1}^{2}}:Y\rightarrow Z_{1}^{2}$ be the projection operator onto $Z_{1}^{2}$ along $Y_{1}$. By $Y_{1}^{2}$ we mean $Y_{1}^{2}=\text {cl Im } P_{1}^{2} F''(x^{*}) h_{1}$.

More generally, define inductively $Y_{k}^{2}:=\text {cl Im } P_{k}^{2} F''(x^{*}) h_{k}$, $k=2,\ldots ,q-1$, where $Z_{k}^{2}$ is a chosen closed complementarity subspace for $(Y_{1}\oplus Y_{1}^{2}\oplus \cdots \oplus Y^{2}_{k-1})$ with respect to Y, and $K_{k}^{2}:= P_{z_{k}^{2}}:Y\rightarrow Z_{k}^{2}$ is the projection operator onto $Z_{k}^{2}$ along $(Y_{1}\oplus Y_{1}^{2}\oplus \cdots \oplus Y^{2}_{k-1})$ with respect to Y, $k=2,\ldots ,q$. Finally $Y_{q}^{2}=Z_{q}^{2}$.

The order q is chosen as the smallest number for which condition (3.1) holds.

Let us define the mappings

$$\begin{aligned} \begin{array}{l} F_{1}:X\rightarrow Y_{1},\ \ F_{1}(x):=P_{Y_{1}}\cdot F(x),\\ F_{2,k}:X\rightarrow Y_{k}^{2},\ \ F_{2,k}(x):=P_{Y_{k}^{2}}\cdot F(x),\ k=1,\ldots ,q, \end{array} \end{aligned}$$

where $P_{Y_{k}^{2}}:Y\rightarrow Y_{k}^{2}$ is the projection operator onto $Y_{k}^{2}$ along

$$\begin{aligned} \left( Y_{1}\oplus Y_{1}^{2}\oplus \cdots \oplus Y^{2}_{k-1}\oplus Y^{2}_{k+1} \oplus \cdots \oplus Y^{2}_{q}\right) . \end{aligned}$$

Definition 3.2

The linear operator

$$\begin{aligned} \Psi _{q}^{2}(x^{*};h_{1},\ldots ,h_{q}))\in \mathcal{L}\left( X,Y_{1}\oplus Y_{1}^{2}\oplus \cdots \oplus Y^{2}_{q}\right) \end{aligned}$$

$h_{k}\ne 0,$ $k=1,\ldots ,q$

$$\begin{aligned} \Psi _{q}^{2}(x^{*};h_{1},\ldots ,h_{q}))=F'_{1}(x^{*})+ F''_{2,1}(x^{*})h_{1}+\ldots F''_{2.q}h_{q} \end{aligned}$$

(3.2)

is called the modified 2-factor-operator.

Definition 3.3

We say that the mapping F is modified 2- regular at $x^{*}$ along $h_{1},h_{2},\ldots ,h_{q}$ if $\text {Im } \Psi _{q}^{2}(x^{*};h_{1},\ldots ,h_{q})=Y$.

Example 3.4

(Continuation of Example 3.1)

Let $h_{1}:=(1,0,0,0)^{T}$, $q=2$, $h_{2}:=(0,0,1,0)^{T}$. Then $P_{Y_{1}}=\left( \begin{array}{ll}0&{}0\\ 0&{}0\end{array}\right)$, $Y_{1}:=\left( \begin{array}{l}0\\ 0\end{array}\right)$, $Y_{1}^{2}:=\left( \begin{array}{l}R\\ 0\end{array}\right)$, $P_{Y_{1}^{2}}=\left( \begin{array}{ll}1&{}0\\ 0&{}0\end{array}\right)$, $Y_{2}^{2}:=\left( \begin{array}{l}0\\ R\end{array}\right)$, $P_{Y_{2}^{2}}=\left( \begin{array}{ll}0&{}0\\ 0&{}1\end{array}\right)$, $F_{1}(x)=0$, $F_{2,1}(x): =\left( \begin{array}{c}x_{1}x_{2}-x_{3}^{2}\\ 0\end{array}\right)$, $F_{2,2}(x):=\left( \begin{array}{l}0\\ x_{3}x_{4}\end{array}\right)$, $F''_{2,1}(0)h_{1}=\left( \begin{array}{llll}0&{}1&{}0&{}0\\ 0&{}0&{}0&{}0\end{array}\right)$, $F''_{2,2}(0)h_{2}=\left( \begin{array}{llll}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}1\end{array}\right)$, and $\Psi _{2}^{2}(0;h_{1},h_{2}):=\left( \begin{array}{llll}0&{}1&{}0&{}0\\ 0&{}0&{}0&{}1\end{array}\right)$ is nonsingular. This means that F(x) is modified 2-regular along $h_{1},$ $h_{2}$.

The following theorem gives the description of elements of the tangent cone $TM(x^{*})$ for modified 2-regular mappings.

Theorem 3.5

Let X, Y be Banach spaces. Let $F:X\rightarrow Y$, $F\in C^{3}(X)$ and $F(x^{*})=0$. Assume that F is modified 2-regular at $x^{*}$ along $h_{1},\ldots ,h_{q}$ and

$$\begin{aligned} \begin{array}{l} F'(x^{*})h_{k}=0,\ k=1,\ldots ,q\\ F''_{2,k}(x^{*})[h_{k+r}, h_{k+p}]=0,\ 1\le k\le q,\ 0\le r\le (q-k), 0\le p\le (q-k-r). \end{array} \end{aligned}$$

(3.3)

Then $h_{1}\in TM(x^{*})$ and there exists $\omega (t)$, $\Vert \omega (t)\Vert =o(t^{1+(q-1)\gamma })$, $\gamma =\frac{1}{2q}$ such that

$$\begin{aligned} F(x^{*}+th_{1}+t^{1+\gamma }h_{2}+\cdots +t^{1+(q-1)\gamma }h_{q}+\omega (t))=0 \end{aligned}$$

and

$$\begin{aligned} \Vert \omega (t)\Vert \le \begin{array}[t]{l} c\Big (\Vert F_{1}(x^{*}+th_{1}+t^{1+\gamma }h_{2}+\cdots +t^{1+(q-1)\gamma }h_{q})\Vert \\ +\sum _{k=1}^{q}\Vert F_{2,k}(x^{*}+th_{1}+t^{1+\gamma }h_{2}+\cdots +t^{1+(q-1)\gamma }h_{q})\Vert ^{\frac{1}{2+(k-1)\gamma }}\Big ), \end{array} \end{aligned}$$

where $c>0$ is a constant.

Example 3.6

(Continuation of Example 3.1)

We show that for Example 3.1 all conditions of Theorem 3.5 are fulfilled. Surjectivity of the operator $\Psi _{2}^{2}(0;h_{1},h_{2})$ along $h_{1}$ and $h_{2}$ have been already shown above.

Now we substantiate condition (3.3). In fact, $F''_{2,1}(0)[h_{1}]^{2}=0$, $F''_{2,2}(0)[h_{2}]^{2}=0$, $F_{2,1}(0)[h_{1},h_{2}]=0$, i.e., all the assumptions of Theorem 3.5 are fulfilled.

For the proof of Theorem 3.5 we need the Multivalued Contraction Mapping Principle (MCPP) proved in Brezhneva and Tretyakov (2007).

Theorem 3.7

(Brezhneva and Tretyakov 2007) For a Banach space $\Omega$ and $w_{0}\in \Omega$, let $\Lambda : B_{r_{1}}(w_{0})\rightarrow 2^{\Omega }$ be a multivalued mapping defined on some ball $B_{r_{1}}(w_{0})\subset \Omega$. Assume that $\Lambda (w)\ne \emptyset$ for any $w\in B_{r_{1}}(w_{0})$ and there exists a number $\alpha \in (0,1)$ such that

• 1. $H(\Lambda (w_{1}),\Lambda (w_{2}))\le \alpha \Vert w_{1}-w_{2}\Vert$ for all $w_{1},w_{2}\in B_{r_{1}}(w_{0})$

• 2. $\text {dist} (w_{0},\Lambda (w_{0}))<(1-\alpha )r_{1}.$

Then for any $r_{2}$ such that $\text {dist} (w_{0},\Lambda (w_{0})<r_{2}<(1-\alpha )r_{1}$ there exists $\bar{w}\in B_{r_{3}}(w_{0})$ with $r_{3}=r_{2}/(1-\alpha )$ such that

$$\begin{aligned} \bar{w}\in \Omega (\bar{w}). \end{aligned}$$

(3.4)

Moreover, among the points $\bar{w}$ satisfying (3.4) there exists a point such that

$$\begin{aligned} \Vert \bar{w}-w_{0}\Vert \le \frac{2}{1-\alpha }\text {dist}(w_{0},\Lambda (w_{0})). \end{aligned}$$

Here $H(\Lambda _{1},\Lambda _{2})$ is the Hausdorff distance between sets $\Lambda _{1}$ and $\Lambda _{2}$,

$$\begin{aligned} H(\Lambda _{1},\Lambda _{2}):=\max \{\sup _{x\in \Lambda _{1}}\text {dist}(x,\Lambda _{2}), \sup _{y\in \Lambda _{2}}\text {dist}(y,\Lambda _{1})\}. \end{aligned}$$

Proof of Theorem 3.5

For the sake of simplicity consider the case $F'(x^{*})=0$. Set $\gamma :=\frac{1}{2}q$ and introduce the following operators

$$\begin{aligned} A_{k}:=F''_{2,k}(x^{*})[h_{k}],\ k=1,\ldots ,q \end{aligned}$$

and

$$\begin{aligned} A:=(tA_{1},\ldots ,t^{1+(q-1)\gamma }Aq). \end{aligned}$$

Define $h(t):=th_{1}+\cdots +t^{1+(q-1)\gamma }h_{q}$ and consider the mapping

$$\begin{aligned} \Lambda (x):=x- A^{-1}(F_{2,1}(x^{*}+h(t)+x),\ldots , F_{2,q}(x^{*}+h(t)+x)). \end{aligned}$$

We show that all assumptions of (MCMP) are satisfied for $\Lambda (x)$ with some ball $U_{r(t)}(0)$, where $r_{1}:=r(t)=o(t^{1+(q-1)\gamma })$.

We start by checking the assumption 2. of (MCMP). By the definition of $\Lambda (0)$, there exists $c_{1}\ge 0$ such that

$$\begin{aligned} \Vert \Lambda (0)\Vert \le c_{1}\Vert A^{-1}F(x^{*}+h(t))\Vert , \end{aligned}$$

where

$$\begin{aligned} \Vert \Lambda (0)\Vert :=\inf \{\Vert x\Vert \ |\ x\in \Lambda (0)\}. \end{aligned}$$

Equivalently,

$$\begin{aligned} \Vert \Lambda (0)\Vert \le c_{2}\left( \frac{\Vert F_{2,1}(x^{*}+h(t))\Vert }{t},\ldots ,\frac{\Vert F_{2,q}(x^{*}+h(t))\Vert }{t^{1+(q-1)\gamma }}\right) . \end{aligned}$$

(3.5)

By using Taylor’s expansion we get for $k=1,\ldots ,q$

$$\begin{aligned} F_{2,k}(x^{*}+h(t))=F''_{2,k}(x^{*})[th_{1}+\cdots +t^{1+(q-1)\gamma }h_{q}]^{2}+\omega _{k}(t), \end{aligned}$$

(3.6)

where $\Vert \omega _{k}(t)\Vert \le c t^{3}$. By definition of mapping $F_{2,k}$, we have

$$\begin{aligned} F''_{2,k}(x^{*})[h_{i},h_{j}]=0, \ \text {for}\ \i <k\ \text {and}\ \ j\le k. \end{aligned}$$

(3.7)

By (3.7) and (3.6), we obtain

$$\begin{aligned} \frac{\Vert F_{2,k}(x^{*}+h(t))\Vert }{t^{1+(k-1)\gamma }}=o(t^{1+(q-1)\gamma })\ \ \text {for}\ k=1,\ldots ,q. \end{aligned}$$

Then, by (3.5) and the latter relation with $\gamma =\frac{1}{2q}$ we obtain

$$\begin{aligned} \Vert \Lambda (0)\Vert =o(t^{1+(q-1)\gamma })=o(t^{1+(q-1)/2q}). \end{aligned}$$

For sufficiently small t with some $\alpha \in (0,1)$ and $r_{1}:=r(t)=o(t^{1+(q-1)\gamma })$ we get

$$\begin{aligned} \Vert \Lambda (0)\Vert <(1-\alpha )r_{1} \end{aligned}$$

which proves assumption 2. of (MCMP).

Now we show that assumption 1. of (MCMP) holds for all $x_{1},x_{2}\in U_{r(t)}(0)$ that is

$$\begin{aligned} H(\Lambda (x_{1}),\Lambda (x_{2}))\le \alpha \Vert x_{1}-x_{2}\Vert ,\ \ 0<\alpha <1. \end{aligned}$$

By the definition of $\Lambda (x)$, with $\bar{x}(t):=x^{*}+h(t)$, we have

$$\begin{aligned} H(\Lambda (x_{1}),\Lambda (x_{2}))=\begin{array}[t]{l} \inf \{\Vert z_{1}-z_{2}\Vert \ |\ z_{i}\in \Lambda (x_{i}),\ i=1,2\}\\ =\inf \{\Vert z_{1}-z_{2}\Vert \ |\ Az_{i}=Ax_{i}-F(\bar{x}(t)+x_{i}),\ i=1,2\}\\ =\inf \{\Vert z_{1}-z_{2}\Vert \ |\ Az_{i}=Ax_{i}-F(\bar{x}(t)+x_{i}),\ i=1,2\}\\ =\inf \{\Vert z\Vert \ |\ Az=A(x_{1}-x_{2})-F(\bar{x}(t)+x_{1})+F(\bar{x}(t)+x_{2})\}\\ \le \Vert A^{-1}(A(x_{1}-x_{2})-F(\bar{x}(t)+x_{1})+F(\bar{x}(t)+x_{2}))\Vert . \end{array} \end{aligned}$$

From this we deduce that

$$\begin{aligned} H(\Lambda (x_{1}),\Lambda (x_{2}))\begin{array}[t]{l} \le c\left( \frac{\Vert F_{2,1}''(x^{*})[th_{1}](x_{1}-x_{2})-F_{2,1}(\bar{x}(t)+x_{1})+F_{2,1}(\bar{x}(t)+x_{2})}{t}\right. \\ +\cdots + \left. \frac{\Vert F_{2,q}''(x^{*})[t^{1+\gamma (q-1)}h_{q}](x_{1}-x_{2})-F_{2,q}(\bar{x}(t)+x_{1})+F_{2,q}(\bar{x}(t)+x_{2})}{t^{1+\gamma (q-1)}}\right) . \end{array} \end{aligned}$$

By using Mean Value Theorem and Taylor’s expansion for $k=1,\ldots ,q$, there exists $c_{k}>0$ such that

$$\begin{aligned} \begin{array}{l} \frac{\Vert F_{2,k}''(x^{*})[t^{1+\gamma (k-1)}h_{k}](x_{1}-x_{2})-F_{2,k}(\bar{x}(t)+x_{1})+F_{2,k}(\bar{x}(t)+x_{2})\Vert }{t^{1+\gamma (k-1)}}\\ \\ \le c_{k}\sup _{\xi \in U_{r_{1}}(0)}\Vert \xi \Vert \frac{\Vert x_{1}-x_{2}\Vert }{t^{1+\gamma (k-1)}}. \end{array} \end{aligned}$$

Hence, with $r_{1}:=r(t)=0(t^{1+\gamma (q-1)})$ we get

$$\begin{aligned} H(\Lambda (x_{1}),\Lambda (x_{2}))\le \alpha \Vert x_{1}-x_{2}\Vert \end{aligned}$$

which proves assumption 1. of (MCMP).

By (MCMP), there exists $\omega (t)$ such that $\omega (t)\in \Lambda (\omega (t))$ which is equivalent to

$$\begin{aligned} 0\in A^{-1}(F(x^{*}+h(t)+\omega (t))). \end{aligned}$$

Hence,

$$\begin{aligned} F(x^{*}+h(t)+\omega (t))=0 \end{aligned}$$

with $\Vert \omega (t)\Vert \le c \Vert \Lambda (0)\Vert =o(t^{1+(q-1)/2q}).$ One can easily show that, by (3.5), and by the inequality $\Vert \omega (t)\Vert \le c\Vert \Lambda (0)\Vert$, we obtain the following estimate

$$\begin{aligned} \Vert \omega (t)\Vert \le c\left( \Vert F_{1}(x^{*}+h(t))\Vert +\sum _{k=1}^{q}\Vert F_{2,k}(x^{*}+h(t))\Vert ^{\frac{1}{2+\gamma (k-1)}}\right) \end{aligned}$$

which finishes the proof. $\square$

3.2 Case $p\ge 3$

We seek an element $x(t)\in M(x^{*})$ in the form

$$\begin{aligned} x(t):=x^{*}+th_{1}+t^{1+\alpha _{2}}h_{2}+\cdots +t^{1+\alpha _{q}}h_{q}+\omega (t) \end{aligned}$$

where $\Vert \omega (t)\Vert =o(t^{1+\alpha _{q}})$, $0<\alpha _{2}<\cdots<\alpha _{q}<1$.

For the sake of simplicity we assume that

$$\begin{aligned} F^{(k)}(x^{*})=0,\ \ k=1,\ldots ,p-1, \ \ q=2. \end{aligned}$$

As previously,

$$\begin{aligned} Y=Y_{1}\oplus \cdots \oplus Y_{p} \end{aligned}$$

where $Y_{1}=\text {cl Im } F^{)p)}(x^{*})[h_{1}]^{p-1},$ $Z_{1}:=Y_{1}$, $Z_{2}$ is closed complementary subspace to $Y_{1}$ and $P_{Z_{1}}:Y\rightarrow Z_{1}$, $P_{Z_{2}}:Y\rightarrow Z_{2}$ are projection operators onto $Z_{1}$ and $Z_{2}$, respectively, $Y_{2}:=\text {cl Im } P_{Z_{2}} F^{(p)}(x^{*})[h_{1}]^{p-2}[h_{2}].$

More generally, we define inductively

$$\begin{aligned} Y_{k}:=\text {cl Im } P_{Z_{k}} F^{(p)}(x^{*})[h_{1}]^{p-k}[h_{2}]^{k-1},\ k=3,\ldots ,p \end{aligned}$$

where $P_{Z_{k}}:Y\rightarrow Z_{k}$ is the projection operator onto $Z_{k}$ along $(Y_{1}\oplus \cdots \oplus Y_{k-1})$ with respect to Y, $k=3,\ldots ,p$. Finally $Y_{p}:=Z_{p}$.

Let us define $F_{k}(x):=P_{k}F(x)$, where $P_{k}:Y\rightarrow Y_{k}$ is the projection operator onto $Y_{k}$ along

$$\begin{aligned} (Y_{1}\oplus \cdots \oplus Y_{k-1}\oplus Y_{k+1}\oplus \cdots \oplus Y_{p}),\ k=1,\ldots ,p. \end{aligned}$$

Definition 3.8

The linear operator $\Psi _{p}(x^{*};h_{1},h_{2})\in \mathcal{L}(X,Y_{1}\oplus \cdots \oplus Y_{p})$ defined as

$$\begin{aligned} \Psi _{p}(x^{*};h_{1},h_{2}){:=}F_{1}^{(p)}(x^{*})[h_{1}]^{p-1}+ F_{2}^{(p)}(x^{*})[h_{1}]^{p-2}[h_{2}]+\cdots +F_{p}^{(p)}(x^{*})[h_{2}]^{p-1} \end{aligned}$$

is called the modified p-factor operator.

Remark 3.9

In case $q\ge 3$ the modified p-factor operator $\Psi _{p}(x^{*};h_{1},\ldots ,h_{q})$ has the following form

$$\begin{aligned} \Psi _{p}(x^{*};h_{1},\ldots ,h_{q}):=\sum _{k=1}^{N(p,q)} F_{k}^{(p)}(x^{*})[h_{i_{1}}]^{q_{1}},\ldots , [h_{i_{p-1}}]^{q_{p-1}}, \end{aligned}$$

where $N(p,q):=\left( \begin{array}{l}q-1\\ p+q-1\end{array}\right)$, $i_{j}\in \{1,\ldots ,p\}$, $i_{k}\ne i_{j}$, for $k\ne j$ and $q_{k}\in \{0,\ldots ,p-1\}$ such that $q_{1}+\ldots ,q_{p-1}=p-1$ and mappings $F_{k}$ are defined in the same way as in case $q=2$ and, obviously, depend on $(p-1)-$ tuple $q_{1},\ldots ,q_{p-1}$.

Definition 3.10

The mapping F is modified p-regular at $x^{*}$ along $h_{1},h_{2}$ if $\text {Im } \Psi _{p}(x^{*};h_{1},h_{2})=Y$ (or $Y_{1}\oplus \cdots \oplus Y_{p}$).

Now we seek an element $x(t)\in M(x^{*})$ in the form

$$\begin{aligned} x(t):=x^{*}+th_{1}+t^{1+\alpha }h_{2}+\omega (t), \end{aligned}$$

(3.8)

where $\alpha :=\frac{1}{2p}$ and $\Vert \omega (t)\Vert =o(t^{1+\alpha })$. The proof of the theorem below remains the same when $\alpha$ assumes any value from a given interval $\alpha \in (0,\varepsilon )$, $\varepsilon <1$.

Theorem 3.11

Let X, Y be Banach spaces. Let $F:X\rightarrow Y$, $F\in C^{p+1}(X),$ $F(x^{*})=0$, $F^{(k)}(x^{*})=0$, $k=1,\ldots ,p-1$.

Assume that F is modified p-regular at $x^{*}$ along $h_{1},h_{2}$ and for the linear operator $\Psi _{p}(x^{*};h_{1},h_{2})$

$$\begin{aligned} \Psi _{p}(x^{*};h_{1},h_{2})\cdot h_{1}=0,\ \ \Psi _{p}(x^{*};h_{1},h_{2})\cdot h_{2}=0. \end{aligned}$$

Then $h_{1}\in TM(x^{*})$ and there exists $\omega (t)$, $\Vert \omega (t)\Vert =o(t^{1+\alpha })$ such that

$$\begin{aligned} F(x^{*}+th_{1}+t^{1+\alpha }h_{2}+\omega (t))=0 \end{aligned}$$

and

$$\begin{aligned} \Vert \omega (t)\Vert \le c \sum _{k=1}^{p}\Vert F_{k}(x^{*}+th_{1}+t^{1+\alpha }h_{2})\Vert ^{\frac{1+\alpha }{p+\alpha k}}, \end{aligned}$$

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. $\square$

4 Degenerate optimization problems

We consider the nonlinear optimization problem

$$\begin{aligned} \begin{array}{l} \min \varphi (x)\\ \text {subject to}\\ F(x)=0, \end{array} \end{aligned}$$

(4.1)

where $\varphi :X\rightarrow \mathbb {R}$ is a sufficiently smooth function and $F:X\rightarrow Y$ 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

$$\begin{aligned} L_{p}(x,\lambda ,h):=\varphi (x)+\left\langle \sum _{K=1}^{p}F_{k}^{(k-1)}(x)[h]^{k-1},\lambda \right\rangle , \end{aligned}$$

where $\lambda \in Y^{*}$, $F_{1}^{(0)}(x):=F(x)$.

The following optimality conditions for p-regular and strongly p-regular mappings F were proved in Brezhneva and Tretyakov (2003).

Theorem 4.1

Let X and Y be Banach spaces. Let $\varphi :X\rightarrow \mathbb {R}$, $\varphi \in C^{2}(X)$, $F:X\rightarrow Y$, $F\in C^{p+1}(X)$. Suppose that $h\in H_{p}(x^{*})$ and F is p-regular along h at the point $x^{*}$.

If $x^{*}$ is a solution to problem (4.1), then there exist multipliers $\lambda ^{*}(h)\in Y^{*}=Y_{1}^{*}\times Y_{2}^{*},\ldots \times Y_{p}^{*}$ such that

$$\begin{aligned} L_{p}'(x^{*},\lambda ^{*}(h),h)=0\ \Leftrightarrow \ \varphi '(x^{*})+ \left( F_{1}'(x^{*})+\cdots +F_{p}^{(p)}(x^{*}[h]^{p-1}\right) ^{*}\lambda ^{*}(h)=0 \end{aligned}$$

(4.2)

Assume that F is strongly p-regular at $x^{*}$.

If there exists $\alpha >0$ and multipliers $\lambda ^{*}(h)\in Y_{1}^{*}\times Y_{2}^{*},\ldots \times Y_{p}^{*}$ such that

$$\begin{aligned} L_{p}'(x^{*},\lambda ^{*}(h),h)=0 \end{aligned}$$

and

$$\begin{aligned} L_{p}''(x^{*}, \bar{\lambda }^{*}(h),h)[h]^{2}\ge \alpha \Vert h_{p}\Vert ^{2}\ \ \forall \ h_{p}\in H_{p}(x^{*}), \end{aligned}$$

where $\bar{\lambda }^{*}(h):=(\lambda ^{*}_{1}(h), 1/3\lambda ^{*}_{2}(h),\ldots ,2/p(p+1)\lambda ^{*}_{p}(h))$, then $x^{*}$ is a strict local mnimizer of problem 4.1.

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

$$\begin{aligned} \min x_{3} \end{aligned}$$

(4.3)

subject to

$$\begin{aligned} F(x):=\left( \begin{array}{c} x_{1}x_{2}-x_{3}^{2}\\ x_{3}x_{4}\end{array}\right) =0. \end{aligned}$$

(4.4)

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

$$\begin{aligned} L_{2,q}(x,\lambda ,h_{1},\ldots ,h_{q}):=\varphi (x)+\left\langle F(x)+\sum _{k=1}^{q}F_{2,k}'(x)h_{k},\lambda \right\rangle . \end{aligned}$$

Theorem 4.3

(Case $p=2$) Let X and Y be Banach spaces, $F:X\rightarrow Y$, $\varphi :X\rightarrow \mathbb {R}$, $\varphi \in C^{2}(X)$ and $F\in C^{3}(X)$.

Assume that there exist elements $h_{1},\ldots ,h_{q}\in X$ such that the mapping F is modified 2-regular at $x^{*}$ along $h_{1},\ldots ,h_{q}$ and assumption 3.3 of Theorem 3.5 is fulfilled.

Then there exists a multiplier $\lambda ^{*}\in Y^{*}$ such that

$$\begin{aligned}&L'_{2,q}(x^{*},\lambda ^{*},h_{1},\ldots ,h_{q})=0\ \Leftrightarrow \ \varphi '(x^{*})\nonumber \\&\quad +\left( F'(x^{*})+F''_{2,1}(x^{*})h_{1}+\cdots +\,F_{2,q}''(x^{*})h_{q}\right) ^{*}\lambda ^{*}=0 \end{aligned}$$

(4.5)

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.

Let us note that for $h_{1}:=(1,0,0,0,)$ and $h_{2}:=(0,0,1,0)$ the mapping F from Example 4.2 defined by (4.4) is modified 2-regular along $h_{1}$ and $h_{2}$ at $x^{*}=0$ and

$$\begin{aligned} \Psi _{q}^{2}(0;h_{1},h_{2})=\left( \begin{array}{llll} 0&{}\quad 1&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 1\end{array}\right) . \end{aligned}$$

If $x^{*}=(0,0,0,0)^{T}$ would solve the problem (4.3), (4.4), then according to Theorem 4.3, there would be a multiplier $\lambda ^{*}\in \mathbb {R}^{2}$ such that

$$\begin{aligned} \left( \begin{array}{l}0\\ 0\\ 1\\ 0\end{array}\right) + \left( \begin{array}{ll}0&{}\quad 0\\ 1&{}\quad 0\\ 0&{}\quad 0\\ 0&{}\quad 1\end{array}\right) \lambda ^{*}=0 \end{aligned}$$

which is impossible.

Consider the case $p\ge 3$ and $q=2$ when $F'(x^{*})=0,\ldots ,F^{(p-1)}(x^{*})=0$. Let us introduce the modified p-factor Lagrange function

$$\begin{aligned} L_{p,2}(x,\lambda ,h_{1},h_{2}):=\varphi (x)+ \left\langle \sum _{k=1}^{p}F_{k}^{(p-1)}(x)[h_{1}]^{p-k}[h_{2}]^{k-1},\lambda \right\rangle . \end{aligned}$$

Theorem 4.4

(Case $p\ge 3$) Let X and Y be Banach spaces, $F:X\rightarrow Y$, $\varphi :X\rightarrow \mathbb {R}$, $\varphi \in C^{2}(X)$ and $F\in C^{p+1}(X)$, and let $x^{*}$ be a solution to optimization problem (4.1).

Assume that there exist elements $h_{1},h_{2}\in X$ such that the mapping F is modified p-regular at $x^{*}$ along $h_{1},h_{2}$ and

$$\begin{aligned} \Psi _{p}(x^{*}; h_{1},h_{2})\cdot h_{k}=0,\ \ k=1,2. \end{aligned}$$

Then there exists a multiplier $\lambda ^{*}\in Y^{*}$ such that

$$\begin{aligned}&L'_{p,2}(x^{*},\lambda ^{*},h_{1},h_{2})=0\ \Leftrightarrow \ \nonumber \\&\quad \varphi '(x^{*})+\left( F_{1}^{(p)}(x^{*})[h_{1}]^{p-1}+F_{2}^{(p)}(x^{*})[h_{1}]^{p-2}[h_{2}]+\cdots \nonumber \right. \\&\qquad \left. +\,F_{p}^{(p)}(x^{*})[h_{2}]^{p-1}\right) ^{*}\lambda ^{*}=0 \end{aligned}$$

(4.6)

Proof

We will show that for any $z\in \text {Ker }\Psi (x^{*};h_{1},h_{2})$ the equality $\langle \varphi '(x^{*}),z\rangle =0$ holds. By annihilator lemmas [ATF], it means that

$$\begin{aligned}&\varphi '(x^{*})\in \text {Im}\Psi (x^{*};h_{1},h_{2})=\text {Im} \left( F_{1}^{(p)}(x^{*})[h_{1}]^{p-1}+F_{2}^{(p)}(x^{*})[h_{1}]^{p-2}[h_{2}]+\cdots \right. \nonumber \\&\quad \left. +\,F_{p}^{(p)}(x^{*})[h_{2}]^{p-1}\right) ^{*} \end{aligned}$$

or, there exists $\lambda ^{*}\in Y^{*}$ such that

$$\begin{aligned}&\varphi '(x^{*})+ \left( F_{1}^{(p)}(x^{*})[h_{1}]^{p-1}+F_{2}^{(p)}(x^{*})[h_{1}]^{p-2}[h_{2}]+\cdots \right. \nonumber \\&\left. \quad +\,F_{p}^{(p)}(x^{*})[h_{2}]^{p-1}\right) ^{*}\lambda ^{*}=0 \end{aligned}$$

Let $z\in \text {Ker }\Psi (x^{*};h_{1},h_{2})$. It means that

$$\begin{aligned}&\left( F_{1}^{(p)}(x^{*})[h_{1}]^{p-1}+F_{2}^{(p)}(x^{*})[h_{1}]^{p-2}[h_{2}]+\cdots \right. \nonumber \\&\left. \quad +\,F_{p}^{(p)}(x^{*})[h_{2}]^{p-1}\right) z=0. \end{aligned}$$

Taking into account the last inequality we will show that there exists $\omega (z,t)$ such that

$$\begin{aligned} x(z,t)=x^{*}+th_{1}+t^{1+\alpha }h_{2}+t^{1+_\alpha +\varepsilon }z+\omega (z,t)\in M(x^{*}), \end{aligned}$$

$t\in [0,\delta ]$, $\Vert \omega (z,t)\Vert =o(t^{1+\alpha +\varepsilon })$, $\alpha +\varepsilon <1$ and $\delta >0$ sufficiently small. To this aim we introduce mapping

$$\begin{aligned} \Lambda _{p}(x):\begin{array}[t]{l} =x-\Psi (x^{*};t h_{1},t^{1+\alpha }h_{2})^{-1} (F_{1}(x^{*}+th_{1}+t^{1+\alpha }h_{2}+t^{1+_\alpha +\varepsilon }z+x)+\cdots \\ +\,F_{p}(x^{*}+th_{1}+t^{1+\alpha }h_{2}+t^{1+_\alpha +\varepsilon }z+x)). \end{array} \end{aligned}$$

Based on the fact that

$$\begin{aligned} \frac{\Vert F_{k}(x^{*}+th_{1}+t^{1+\alpha }h_{2}+t^{1+_\alpha +\varepsilon }z)\Vert }{t^{p-k}t^{(1-\alpha )(k-1)}}=O(t^{2}) \end{aligned}$$

for $k=1,\ldots ,p$ we obtain, analogously as in the proof of Theorem 3.5, that mapping $\Lambda _{p}(x)$ satisfies all the assumptions of (MCMP) with some ball $B_{\bar{r}(t)}(0)$, where $r_{1}:=\bar{r}(t)=o(t^{1+\alpha +\varepsilon })$.

By (MCMP), there exists $\omega (z,t)\in \Lambda _{p}(\omega (z,t))$ which is equivalent to

$$\begin{aligned} 0\in \Psi (x^{*};th_{1},t^{1+\alpha }h_{2})^{-1}F(x^{*}+th_{1}+t^{1+\alpha }h_{2}+t^{1+_\alpha +\varepsilon }z+\omega (z,t)), \end{aligned}$$

$t\in [0,\delta ]$ with

$$\begin{aligned} \Vert \omega (z,t)\Vert \le c\Vert \Lambda _{p}(0)\Vert =o(t^{1+\alpha +\varepsilon }). \end{aligned}$$

It means that

$$\begin{aligned} x(z,t):=x^{*}+th_{1}+t^{1+\alpha }h_{2}+t^{1+_\alpha +\varepsilon }z+\omega (z,t)\in M(x^{*}) \end{aligned}$$

for all $z\in \text {Ker }\Psi _{p}(x^{*};h_{1},h_{2})$.

Now we finish the proof by observing that it must be

$$\begin{aligned} \langle \varphi '(x^{*}),h_{1}\rangle = \langle \varphi '(x^{*}),h_{2}\rangle = \langle \varphi '(x^{*}),z\rangle =0 \end{aligned}$$

for all $z\in \text { Ker }\Psi _{p}(x^{*};h_{1},h_{2})$ since otherwise $x^{*}$ is not a minimizer of our problem. $\square$

5 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 $TM(x^{*})$ to the set $M(x^{*})$ at the point $x^{*}$ of the mapping F but knowing a single element $h\in TM(x^{*})$ 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.