Implicit function and tangent cone theorems for singular inclusions and applications to nonlinear programming
Abstract
The paper is devoted to the implicit function theorem involving singular mappings. We also discuss the form of the tangent cone to the solution set of the generalized equations in singular case and give some examples of applications to nonlinear programming and complementarity problems.
Keywords
Implicit function theorem Multifunction Tangent cone Singularity Generalized equation pRegularity1 Introduction
We propose another approach to implicit function theorem for generalized equation (2) that differs from the results of the above mentioned papers, i.e. we focus on singular mappings. We do not assume surjectivity of the derivative of the mapping f. Such problems (2) are called singular inclusion problems.
More precisely, we investigate the problem of existence of a locally defined mapping \(\varphi : X\rightarrow Y\), \(y=\varphi (x)\) which is a solution of (2) near a given solution \((x_0,y_0)\), that is \(0\in f(x,\varphi (x))+N_C(\varphi (x))\) for x close to \(x_{0}\) and \(y_0=\varphi (x_0)\).
The classical implicit function theorem says that when a continuously differentiable function f(x, y) vanishes at a point \((x_0,y_0)\) with \(f'_y(x_0,y_0)\) nonsingular (surjective), the equation \(f(x,y)=0\) can be solved for y in terms of x in a neighborhood of \((x_0,y_0)\). This theorem has been extended in various directions, e.g. to Banach spaces [25, 26], to multivalued mappings [2, 10, 11, 16, 22], to nonsmooth functions [16, 23], etc.
The results we present can be applied to parametric problems. There are numerous theorems concerning the solution existence of the problem with small parameter. Some of them deal with the problem of solution existence of the equation \(f(x,y)=0\), where the mapping f is singular, e.g. [4, 5, 20, 21, 27]. This analysis was based on the constructions of pregularity theory that has been developed for the last forty years. The main constructions of this theory are described e.g. in [3, 4, 5, 6] or in [25, 26, 27].
The organization of the paper is as follows. In Sect. 2 we formulate our main result which is Theorem 1 providing conditions for the existence of the implicit function \(\varphi \) for the singular inclusion problem (2). In Sect. 3 we prove Lusterniktype theorem for (2).
2 Implicit function theorem for singular inclusions
We prove the following theorem.
Theorem 1
 \(1^{\circ }\)

Banach condition:
for any \(x\in U_{\gamma }(0)\), such that \(f(x,0)\ne 0\) and \(\gamma >0\) is sufficiently small, there exists \(h(x)\in Y\), \(h\ne 0\) such thatand \(\Vert h(x)\Vert \le c\cdot \Vert f(x,0)\Vert ^{1/p}\), where \(c>0\) is independent constant,$$\begin{aligned} f(x,0)\in \frac{1}{(p1)!}f_y^{(p)}(0,0)[h(x)]^p+N_C(h(x)), \end{aligned}$$(6)  \(2^{\circ }\)
 Strong pregularity condition at the point 0 along \(h=h(x)\), \(h\in Y\), i.e.$$\begin{aligned} H(L_h^{1}(z_1),L_h^{1}(z_2))\le \frac{c}{\Vert h\Vert ^{p1}}\Vert z_1z_2\Vert , \quad \forall z_1, z_2 \in Y^{*}, \end{aligned}$$(7)
 \(3^{\circ }\)
 pfactor approximation condition, i.e.for \(x\in U_{\gamma }(0),\)\(y_1\), \(y_2\in V_{\gamma }(0)\) and \(\delta >0\) sufficiently small.$$\begin{aligned}&\left\ f(x,y_1)f(x,y_2)\frac{1}{p!}f_y^{(p)}(0,0)[y_1]^p+\frac{1}{p!}f_y^{(p)}(0,0)[y_2]^p\right\ \nonumber \\&\quad \le \delta \left( \Vert y_1\Vert ^{p1}+\Vert y_2\Vert ^{p1}\right) \Vert y_1y_2\Vert \end{aligned}$$(8)
Remark
In the case when \(f(x,0)=0\) we can take the mapping \(\varphi (x)=0\) and Banach condition is trivial.
Before we prove this theorem we give two examples and for the convenience of the Reader we recall Robinson’s strong regularity condition [22].
Definition 1
Example 1
On the other hand, it turns out that all assumptions of Theorem 1 are fulfilled for \(p=2\).
The second assumption, strong pregularity \(2^{\circ }\) holds since f(x, y) is 2regular (see e.g. in [6]) at \((0,0)^T\) with respect to y along any \(h\in Y\), such that \(h_1\ne 0\) and \(h_2\ne 0\), that is \(\mathrm{Im}\,f''(0,0)h={\mathbb {R}}^2\) and \(N_{{\mathbb {R}}_+^2}(\cdot )=\{0\}\).
The pfactor approximation condition \(3^{\circ }\) is immediately satisfied due to the form of the mapping f(x, y). This means that all conditions of Theorem 1 are fulfilled and therefore for small x there exists a mapping \(\varphi (x)\) which acts from a neighborhood of \(0\in {\mathbb {R}}^2\) into \({\mathbb {R}}^2\) and for which (10) is satisfied.
Example 2
The following theorem is essential in the proof of Theorem 1 (see [15]).
Theorem 2
 1.
\(H(\varPhi (z_{1}),\varPhi (z_{2}))\le \theta \rho (z_{1},z_{2})\) for any \(z_{1},z_{2}\in U_{\varepsilon }(z_{0})\)
 2.
\(\rho (z_{0},\varPhi (z_{0}))<(1\theta ) \varepsilon .\)
Proof
of Theorem 1
Suppose that \(x\in U_{\gamma }(0)\), \(y_1,y_2\in V_{\gamma }(0)\), for sufficiently small \(\gamma >0\). Moreover, assume that there exists \(h(x)\in Y\), \(h\ne 0\) such that Banach condition holds true.
\(0\in N_C(h+y(x))+f(x,h+y(x)).\) For this purpose we check the assumptions of CMP.
 1.
\(H(\varPhi (x, y_1), \varPhi (x,y_2))\le \!\!\!\!\!\!\!^{^{2^{\circ }}} \frac{c}{\Vert h\Vert ^{p1}}\Vert r(x,h+y_1)r(x,h+y_2)\Vert =\frac{c}{\Vert h\Vert ^{p1}}\cdot \)
\(\cdot \Vert \frac{1}{(p1)!}f_y^{(p)}(0,0)[h]^{p1}[y_1y_2]f(x,h+y_1)+f(x,h+y_2)\Vert \)
\(\le \!\!\!\!\!\!^{^{3^{\circ }}}\frac{c\cdot \delta (\Vert h\Vert ^{p1}+\Vert h\Vert ^{p1})}{\Vert h\Vert ^{p1}}\Vert y_1y_2\Vert \le 2 c\cdot \delta \Vert y_1y_2\Vert \).
Moreover, since \( 0\in L_h^{1}\left( f(x,0)\right) \) then
 2.
\(H(\varPhi (x,0), 0)\le H\left( L_h^{1}(r(x,h)),L_h^{1}(f(x,0))\right) \le \)
\(\le \!\!\!\!\!\!^{^{2^{\circ }}} \frac{c}{\Vert h\Vert ^{p1}}\Vert r(x,h)+f(x,0)\Vert \)
\(=\frac{c}{\Vert h\Vert ^{p1}}\left\ \frac{1}{(p1)!}f_y^{(p)}(0,0)[h]^{p}f(x,h)+f(x,0)\right\ \)
\(\le \!\!\!\!\!\!^{^{3^{\circ }}} \frac{c\cdot \delta }{\Vert h\Vert ^{p1}}\Vert h\Vert ^{p1}\Vert h\Vert \le c \delta \Vert h\Vert \).
3 Porder tangent cone theorem for singular inclusions: generalization of Lusternik theorem
Definition 2
It is enough to consider the completely degenerate case up to the order p, i.e. the case where \(f^{(k)}(x_0)=0,\)\(k=1,\ldots ,p1\), \(p\ge 2\).
Theorem 3
Proof
For the sake of simplicity we assume that \(x_0=0\). Let us define
\( r(t{\bar{h}}+x):=\frac{1}{(p1)!}f^{(p)}(0)[t{\bar{h}}]^{p1}[t{\bar{h}}+x]f(t{\bar{h}}+x)\).
 1\(^{\circ }\)

\(H(\varPhi (y_1),\varPhi (y_2))= H\left( L_{t{\bar{h}}}^{1}\left( r(t{\bar{h}}+y_1)\right) ,L_{t{\bar{h}}}^{1}\left( r(t{\bar{h}}+y_2)\right) \right) \)
\(\le \frac{c}{t^{p1}}\left\ \frac{1}{(p1)!}f^{(p)}(0)[t{\bar{h}}]^{p1}(y_1y_2) f(t{\bar{h}}+y_1)+f(t{\bar{h}}+y_2)\right\ \)
\(\le \delta (t)\Vert y_1y_2\Vert ,\) where \(\delta (t)\rightarrow 0\) while \(t\rightarrow 0\).
 2\(^{\circ }\)

\(H(\varPhi (0),0)\le \)
\(\le H\left( L_{t{\bar{h}}}^{1}\left( r(t{\bar{h}})\right) , L_{t{\bar{h}}}^{1}\left( \frac{1}{(p1)!}f^{(p)}(0)[t{\bar{h}}]^{p}+N_C(t{\bar{h}})\right) \right) \)
\(\le \frac{c}{t^{p1}}\left\ r(t{\bar{h}})\frac{1}{(p1)!}f^{(p)}(0)[t{\bar{h}}]^{p}+N_C(t{\bar{h}})\right\ \le \)
\(\le \frac{c}{t^{p1}}\left\ \frac{1}{(p1)!}f^{(p)}(0)[t{\bar{h}}]^{p} f(t{\bar{h}})+0\right\ \le c_1 t^2.\)
It means that all conditions of CMP are fulfilled and hence there exists \(w(t{\bar{h}})\in \varPhi (t{\bar{h}})\) and it follows that \(w(t{\bar{h}})\in L_{t{\bar{h}}}^{1}\left( r(t{\bar{h}}+w(t{\bar{h}}))\right) \), or in other words \( 0\in f(t{\bar{h}}+w(t{\bar{h}}))+N_C(t{\bar{h}}+w(t{\bar{h}})\)\(\forall t\in [0,\varepsilon )\) and \(\Vert w(t{\bar{h}})\Vert =o(t)\), i.e. \({\bar{h}}\in TM_F(x_0)\). \(\square \)
Example 3
Let \(F(x)=f(x)+N_C(x)\), where \(f(x)=(f_1(x),f_2(x))^T\), \(x\in {\mathbb {R}}^2\),
Taking \({\bar{h}}=({\bar{h}}_{x_1},{\bar{h}}_{x_2})^T\) where \({\bar{h}}_{x_1}=0\)\({\bar{h}}_{x_2}=1\) we have \(N_{{\mathbb {R}}_+^2}\left( {\begin{matrix} 0 \\ 1\\ \end{matrix}} \right) = \alpha \left( \begin{array}{c} 1 \\ 0\\ \end{array} \right) \), \(\alpha >0\) and \({\bar{h}}\in \mathrm{Ker}^2L_h(0)\) if \(\left( \begin{array}{c} {\bar{h}}_{x_1} \\ 0\\ \end{array} \right)  \left( \begin{array}{c} \alpha \\ 0\\ \end{array} \right) =0\), i.e. \({\bar{h}}_{x_1}^2=\alpha \) and \({\bar{h}}_{x_1}=\pm \sqrt{\alpha }\) and all the assumptions of Theorem 3 are fulfilled. It means that \({\bar{h}}=(0,1)^T\in TM_F(0).\)
Both theorems presented in the present paper have been proved only for completely degenerate case but they could be easily generalized for noncomplete degeneration. Such generalization follows directly from the construction of the pfactor operator and we refer the Reader to the papers devoted to the pregularity theory (see e.g. [6, 7, 21, 26]).
Notes
Acknowledgements
The results of the research of the second and third author carried out under the research theme No. 165/00/S were financed from the science grant granted by the Ministry of Science and Higher Education. The work of the third author was supported also by the Russian Foundation for Basic Research (project No. 170700510, 170700493) and the RAS Presidium Programme (programme 27).
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