On Lipschitz-like continuity of a class of set-valued mappings

TL;DR

This paper investigates the Lipschitz-like continuity of set-valued mappings derived from parametric systems of equalities and inequalities, establishing conditions under relaxed constraint qualifications.

Contribution

It introduces new conditions for Lipschitz-like continuity of set-valued mappings based on relaxed constant rank constraint qualification.

Findings

Lipschitz-like continuity holds under relaxed conditions

Provides theoretical framework for parametric systems

Extends previous results with weaker assumptions

Abstract

We study set-valued mappings defined by solution sets of parametric systems of equalities and inequalities. We prove Lipschitz-like continuity of these mappings under relaxed constant rank constraint qualification.

Equations

$$\begin{array}[]{ll}\text{minimize}&f(p,x)\\ \text{subject to}&x\in F(p)=\{x\in\mathcal{H}\mid{h_{i}}(p,x)\leq 0,\ i\in I,\ h_{i}(p,x)=0,\ i\in I_{0}\},\end{array}$$

$$T(F(p),x)=\{d\in \mathcal{H}\mid \exists t_k\downarrow 0,d^k\to d\text{ such that }x+t_k{d}^k\in F(p),k=1,2,\dots \},$$

$$\Gamma (F(p),x)=\{d\in \mathcal{H}\mid ⟨{\nabla }_x{h}_i(p,x),d⟩\le 0:i\in I(p,x),⟨{\nabla }_x{h}_i(p,x),d⟩=0:i\in I_0\}.$$

$$\text{dist}(x^0,F(p))\le l\Vert p-p^0\Vert \forall p\in V_{\delta }(p^0)(\forall p\in V_{\delta }(p^0)\cap P).$$

$$F(p^1)\cap V(x^0)\subset F(p^2)+l_F\Vert p^2-p^1\Vert B$$

$$\text{rank}\{{\nabla }_x{h}_i(p,x):i\in I_0\cup K\}=\text{rank}\{{\nabla }_x{h}_i(p^0,x^0):i\in I_0\cup K\}$$

$$\text{rank}\{{\nabla }_x{h}_i(p^0,x):i\in I_0\cup K\}=\text{rank}\{{\nabla }_x{h}_i(p^0,x^0):i\in I_0\cup K\}$$

$$\begin{array}{rl}& \text{dist}(x,F(p))\le M\max \{0,h_i(p,x),i\in I,|h_i(p,x)|,i\in I_0\}\\ & \text{for all}x\in V(x^0)\text{and}p\in V(p^0)(p\in V(p^0)\cap P).\end{array}$$

$$\begin{array}[]{ll}\text{minimize}&{f_{v}}(p,x)=\|{x-v}\|\\ \text{subject to}&x\in F(p).\end{array}$$

$$\begin{array}[]{ll}\text{minimize}&\frac{1}{2}\|{x-v}\|^{2}\\ \text{subject to}&x\in F(p).\end{array}$$

$${\Lambda }_v(p,x):=\{\lambda \in {\mathbb{R}}^n\mid \frac{x-v}{\Vert x-v\Vert }+\sum _{i=1}^n{\lambda }_i{\nabla }_x{h}_i(p,x)=0,{\lambda }_i\ge 0,{\lambda }_i{h}_i(p,u)=0\text{for}i\in I\},$$

$${\Lambda }_v^M(p,x):=\{\lambda \in {\Lambda }_v(p,x)\mid \sum _{i=1}^n|{\lambda }_i|\le M\}.$$

$$\text{dist}(v^k,F(p^k))>k\max \{0,h_i(p^k,v^k),i\in I,|h_i(p^k,v^k)|,i\in I_0\}.$$

$$\begin{array}[]{l}\frac{x^{k}-v^{k}}{\|x^{k}-v^{k}\|}+\sum_{i=1}^{n}\lambda_{i}^{k}\nabla_{x}h_{i}(p^{k},x^{k})=0,\ \lambda_{i}^{k}\nabla_{x}h_{i}(p^{k},x^{k})=0,\\ \lambda_{i}^{k}\geq 0\ i\in I(p^{k},x^{k}),\ \lambda_{i}^{k}=0\ i\in I\setminus I(p^{k},x^{k}).\end{array}$$

$$\Vert x^k-v^k\Vert =⟨\sum _{i=1}^n{\lambda }_i^k{\nabla }_x{h}_i(p^k,x^k),v^k-x^k⟩$$

$$\le \sum _{i=1}^n{\lambda }_i^k(h_i(p^k,v^k)-h_i(p^k,x^k)+o(\Vert v^k-x^k\Vert ))=$$

$$=\sum _{i=1}^n{\lambda }_i^k{h}_i(p^k,v^k)+\sum _{i=1}^n{\lambda }_i^{k}o(\Vert v^k-x^k\Vert ))\le \sum _{i=1}^n{\lambda }_i^k{h}_i(p^k,v^k)+\frac{1}{2}\Vert v^k-x^k\Vert .$$

$$\text{dist}(v^k,F(p^k))=\Vert x^k-v^k\Vert \le 2M\max \{0,h_i(p^k,v^k),i\in I,|h_i(p^k,v^k)|,i\in I_0\},$$

$$\text{rank}\{{\nabla }_x{h}_i(p,x),i\in I_0\}=\text{rank}\{{\nabla }_x{h}_i(p^0,x^0),i\in I_0\}.$$

$$\text{rank}\{{\nabla }_x{h}_i(p,x),i\in I_0^{\prime }\}=\text{rank}\{{\nabla }_x{h}_i(p^0,x^0),i\in I_0^{\prime }\}=|I_0^{\prime }|,$$

$$\text{dist}(0,{\Lambda }_{v_k}(p_k,x_k))\to +\infty .$$

$$\text{rank}\{{\nabla }_x{h}_i(p,x),i\in J\}=\text{rank}\{{\nabla }_x{h}_i(p^0,x^0),i\in J\}\forall (p,x)\in V(p^0,x^0).$$

$$v_k-x_k=\sum _{i\in I_0\cup I(p_k,x_k)}{\hat{\lambda }}_i^k{\nabla }_x{h}_i(p_k,x_k),k=1,\dots$$

$$\frac{v_k-x_k}{\Vert v_k-x_k\Vert }=\sum _{i\in I_0\cup I(p_k,x_k)}{\lambda }_i^k{\nabla }_x{h}_i(p_k,x_k)⟺v_k-x_k=\sum _{i\in I_0\cup I(p_k,x_k)}{\hat{\lambda }}_i^k{\nabla }_x{h}_i(p_k,x_k),$$

$$\sum _{i\in I_0}{\hat{\lambda }}_i^k{\nabla }_x{h}_i(p_k,x_k)=\sum _{i\in I_0^{\prime }}{\check{\lambda }}_i^k{\nabla }_x{h}_i(p_k,x_k),$$

$$v_k-x_k=\sum _{i\in I_0^{\prime }}{\check{\lambda }}_i^k{\nabla }_x{h}_i(p_k,x_k)+\sum _{i\in I(p_k,x_k)}{\hat{\lambda }}_i^k{\nabla }_x{h}_i(p_k,x_k),$$

$$\sum _{i\in I_0^{\prime }}{\check{\lambda }}_i^k{\nabla }_x{h}_i(p_k,x_k)+\sum _{i\in I^0}{\hat{\lambda }}_i^k{\nabla }_x{h}_i(p_k,x_k)=\sum _{i\in I_0^{\prime }}{\tilde{\lambda }}_i^k{\nabla }_x{h}_i(p_k,x_k)+\sum _{i\in I^k}{\tilde{\lambda }}_i^k{\nabla }_x{h}_i(p_k,x_k),$$

$$v_k-x_k=\sum _{i\in I_0^{\prime }}{\tilde{\lambda }}_i^k{\nabla }_x{h}_i(p_k,x_k)+\sum _{i\in I^k}{\tilde{\lambda }}_i^k{\nabla }_x{h}_i(p_k,x_k).$$

$$\frac{v_k-x_k}{\Vert v_k-x_k\Vert \Vert {\lambda }^k\Vert }=\sum _{i\in I_0^{\prime }}\frac{{\lambda }_i^k}{\Vert {\lambda }_k\Vert }{\nabla }_x{h}_i(p_k,x_k)+\sum _{i\in I^k}\frac{{\lambda }_i^k}{\Vert {\lambda }_k\Vert }{\nabla }_x{h}_i(p_k,x_k).$$

$$0=\sum _{i\in I_0^{\prime }\cup I^{\prime }}{\bar{\lambda }}_i{\nabla }_x{h}_i(p^0,x^0),{\bar{\lambda }}_i\ge 0,i\in I^{\prime },$$

$$\text{rank}\{{\nabla }_x{h}_i(p_0,x_0),i\in I_0^{\prime }\cup I^{\prime }\}$$

$$=\text{rank}\{{\nabla }_x{h}_i(p_0,x_0),i\in I_0\cup I^{\prime }\}$$

$$=\text{rank}\{{\nabla }_x{h}_i(p_k,x_k),i\in I_0\cup I^{\prime }\}$$

Full text

1 Systems Research Institute of the Polish Academy of Sciences, Warsaw University of Technology

2 Belarus State University of Informatics and Radioelectronics, Minsk, Belarus

3 Warsaw University of Technology, Systems Research Institute of the Polish Academy of Sciences

1. Introduction

Properties of set-valued mappings given by systems of equalities and inequalities play a significant role in parametric optimization. In particular, considerable effort is directed towards formulating conditions ensuring Lipschitz-type continuities of these mappings, namely their calmness and the pseudo-Lipschitz continuity (also referred to as Lipschitz-like continuity or the Aubin property) [1, 13, 18, 19, 24].

The present paper is devoted to sufficient conditions for R-regularity (Definition 2.5) and pseudo-Lipschitz continuity for set-valued mappings defined by solution sets of parametric constrained systems. R-regularity is a variant of a much more general property, called metric regularity, intensively studied in [8, 9, 11, 16, 17, 21].

Let $\mathcal{H}$ be a Hilbert space and $\mathcal{G}$ be a normed space. Let us consider a parametric nonlinear programming problem:

[TABLE]

where $p\in\mathcal{G}$ is a parameter, $x\in\mathcal{H}$ stands for the decision variable, $I=\{1,\dots,m\}$, ${I_{0}}=\{m+1,\dots,n\}$ (we admit the case $I_{0}=\emptyset$). Functions $f,{h_{i}}:\ \mathcal{G}\times\mathcal{H}\rightarrow\mathbb{R}$, $i=1,\dots,n$, are assumed to be (jointly) continuous together with their partial gradients with respect to $x$, ${\nabla_{x}}f$ and ${\nabla_{x}}{h_{i}}$, $i=1,\dots,n$.

In the present paper we prove Lipschitz-likeness of the set-valued mapping $F$ defined in (1.1). We generalize results from [25] and [3]. In [25] the respective results are obtained under stronger assumptions of functions $h_{i}$, $i\in I_{0}\cup I$, while in [3] the Lipschitz-likeness of $F$ is obtained for $h_{i}(p,x)=\langle x\mid g_{i}(p)\rangle-f_{i}(p)$, $i\in I_{0}\cup I$, where $f_{i}:\ \mathcal{G}\rightarrow\mathbb{R}$, $g_{i}:\ \mathcal{G}\rightarrow\mathcal{H}$, $i\in I_{0}\cup I$, are locally Lipschitz functions. We also correct the mistake in the proof of Lemma 3 of [25].

For set-valued mapping $F:\ \mathcal{G}\rightrightarrows\mathcal{H}$ defined in (1.1), its domain and graph are defined by $domF=\{p\in\mathcal{G}\mid F(p)\neq\emptyset\}$ and $grF=\{(p,x)|x\in F(p),\ p\in\mathcal{G}\}$, respectively.

The tangent cone $T(F(p),x)$ and the linearized cone $\Gamma(F(p),x)$ to $F(p)$ at $x\in F(p)$ are defined, respectively, as follows

[TABLE]

Organization of the paper is as follows. In section 2 we provide basing concepts. Section 3 is devoted to the concept of $R-$regularity. In section 4 we investigate relationships between relaxed constant rank condition and the R-regularity of $F$. In section 5 we prove Lipschitz-likeness of $F$ under relaxed constant rank condition. In section 6 some applications to bilevel programming are discussed.

2. Basic concepts and definitions

This section contains some background material (see, i.e., [1, 15, 19, 22, 24, 26]) which will be used in the sequel.

We denote ${V_{\delta}}({p^{0}})={p^{0}}+\delta B$, ${V_{\varepsilon}}({x^{0}})={x^{0}}+\varepsilon B$, where $B$ is the open unit ball centered at [math] in the respective space, $\text{dist}(v,C):=\inf\{\|v-c\|,\ c\in C\}$ is the distance between point $v$ and set $C$, where $\|v\|$ is the norm of vector $v$.

Definition 2.1**.**

A set-valued mapping $F$ is lower semicontinuous (lsc) at $({p^{0}},{x^{0}})\in grF$ (relative to $P\subset\mathcal{G}$) if for any neighbourhood $V({x^{0}})$ there is a neighborhood $V({p^{0}})$ such that $F(p)\cap V({x^{0}})\neq\emptyset$ for all $p\in V({p^{0}})$ (for all $p\in V({p^{0}})\cap P$).

Definition 2.2**.**

A set-valued mapping $F$ is lower Lipschitz continuous at $({p^{0}},{x^{0}})\in grF$ (relative to $P\subset\mathcal{G}$) if there exist positive numbers $l$ and $\delta$ such that

[TABLE]

Note that (2.1) implies that $F(p)\cap{V_{\varepsilon}}({x^{0}})\neq\emptyset$ for any $\varepsilon>0$, if $\|{p-{p^{0}}}\|<\min\{\delta,\varepsilon/l\}$.

Definition 2.3**.**

A set-valued mapping $F$ is Lipschitz-like (pseudo-Lipschitzian) (relative to $P\subset\mathcal{G}$) at $({p^{0}},{x^{0}})\in grF$ (where ${p^{0}}\in P$) if there exist a number ${l_{F}}>0$ and neighbourhoods $V({p^{0}})$ and $V({x^{0}})$ such that

[TABLE]

for all ${p^{1}},{p^{2}}\in V({p^{0}})$ (${p^{1}},{p^{2}}\in V({p^{0}})\cap P$).

Let $P\subset\mathcal{G}$, $X\subset\mathcal{H}$ and $I(p,x):=\{i\in I\ |\ h_{i}(p,x)=0\}$ be the set of indices of active inequality constraints at $(p,x)\in grF$ . Following [24, 25] we define the relaxed constant rank constraint qualification (RCRCQ) which generalizes the constant rank constraint qualification introduced by Janin [18].

Definition 2.4**.**

The set-valued mapping $F$ satisfies the Relaxed Constant Rank Constraint Qualification, or shortly, RCRCQ (relative to $P\times X$) at $({p^{0}},{x^{0}})\in grF$, if for any index set $K\subset I(p{{}^{0}},{x^{0}})$

[TABLE]

in a neighbourhood of $(p{{}^{0}},{x^{0}})$ (for $(p,x)\in P\times X$ from this neighbourhood).

The set $F(p^{0})$ satisfies RCRCQ at $x^{0}\in F(p^{0})$ if for any index set $K\subset I(p^{0},x^{0})$

[TABLE]

for all $x$ in a neighbourhood of $x^{0}$.

Clearly, if the set-valued mapping $F$ satisfies RCRCQ at $({p^{0}},{x^{0}})\in grF$, then it satisfies RCRCQ at all points $(p,x)\in gr\,F$ in some neighbourhood of $({p^{0}},{x^{0}})$.

The following lemma proves the equality $\Gamma(F(p),x)=T(F(p),x)$ under RCRCQ. In the finite dimensional case, where $\mathcal{H}=\mathbb{R}^{s}$ this fact has been proved in Theorem 1 of [24]. In the infinite-dimensional case considered in the present paper this fact has been proved in Theorem 6.3 of [2].

Lemma 2.1**.**

([2, 24]). Let the set-valued mapping $F$ satisfy RCRCQ (relative to $domF\times\mathcal{H}$) at $({p^{0}},{x^{0}})\in grF$. Then there exist neighborhoods $V({p^{0}})$ and $V({x^{0}})$ such that $\Gamma(F(p),x)=T(F(p),x)$ for all $x\in F(p)\cap V({x^{0}})$ and all $p\in V({p^{0}})\cap domF$.

Proof.

As already noted, if $F$ satisfies RCRCQ at $(p^{0},x^{0})$, there are neighbourhoods $V({p^{0}})$ and $V({x^{0}})$ such that $F$ satisfies RCRCQ at any point $(p,x)$ where $p\in V({p^{0}})\cap domF$, $x\in F(p)\cap V({x^{0}})$. Hence, the set $F(p)$ satisfies RCRCQ at $x$ and by Theorem 6.3 of [2], $\Gamma(F(p),x)=T(F(p),x)$. ∎

Following [12, 22] we define the R-regularity of set-valued mappings.

Definition 2.5**.**

The set-valued mapping $F$ is R-regular at $({p^{0}},{x^{0}})\in grF$ (relative to $P\subset\mathcal{G}$) if there exist a number $M>0$ and neighbourhoods $V({p^{0}})$ and $V({x^{0}})$ such that

[TABLE]

The concept of R-regularity appears in different works (see e.g. Theorem 2.84 and formula (2.164) of [4], formula (10) of [10]). When $h_{i}(x,p)=g_{i}(x)-p_{i}$, $g_{i}(p):\ \mathcal{G}\rightarrow\mathbb{R}$, $p_{i}\in\mathbb{R}$ $i\in I\cup I_{0}$ the R-regularity is equivalent to the metric regularity of $G(x,p):=[g_{i}(x)-p_{i}]_{i\in I\cup I_{0}}-K$, where $K=\mathbb{R}_{-}^{m}\times\{0\}^{n-m}$ (see formulas (2.143), (2.144) of [4]). In the paper [20], some variants of (2.5) have been investigated (see e.g. formula (6) of [20]).

3. Criterion of R-regularity

Let $v\in\mathcal{H}$ and $\Pi_{F(p)}(v):=\{\tilde{v}\in F(p)\mid\|\tilde{v}-v\|=\text{dist}(v,F(p))\}$. The set $\Pi_{F(p)}(v)$ is the solution set to the problem

[TABLE]

The problem (3.1) can be equivalently reformulated as

[TABLE]

Lagrange multiplier sets for problem (3.1) are defined as follows

[TABLE]

Lemma 3.1**.**

(Proposition 7.1 of [2], Theorem 1 of [24]). Suppose that $x^{0}\in\mathcal{H}$ is a solution to (3.1) for $p=p^{0}\in\mathcal{G}$ and $v\in\mathcal{H}$. Assume the set-valued mapping $F$ satisfies RCRCQ (relative to $domF\times\mathcal{H}$) at $({p^{0}},{x^{0}})\in grF$. Then $\Lambda_{v}(p^{0},x^{0})\neq\emptyset$.

The following theorem generalizes Theorem 2 [25] and Theorem 4.1 [14] to parametric systems defined by the set-valued mapping $F$.

Theorem 3.1**.**

Let $({p^{0}},{x^{0}})\in grF$ and the set-valued mapping $F$ be l.s.c. at $({p^{0}},{x^{0}})$ relative to $domF$. The following assertions are equivalent:

1. (a)

the set-valued mapping $F$ is R-regular at $({p^{0}},{x^{0}})$ relative to $domF$; 2. (b)

there exists a number $M>0$ such that for any sequences ${p^{k}}\to{p^{0}},{p^{k}}\in domF,$ ${v^{k}}\to{x^{0}},{v^{k}}\notin F({p^{k}})$, the inequality $\Lambda_{v_{k}}^{M}({p^{k}},{x^{k}})\neq\emptyset$ holds for all ${x^{k}}=x({p^{k}},{v^{k}})\in\Pi_{F({p^{k}})}({v^{k}})$ and $k$ sufficiently large.

Proof.

If ${x^{0}}\in{\mathop{\rm int}}F({p^{0}})$, the theorem is obviously valid. Assume that ${x^{0}}\in bdF({p^{0}})$.

The implication $(a)\Rightarrow(b)$ follows from a slight modification in the first part of the proof of Theorem 2 [25]. 2. 2)

$(b)\Rightarrow(a)$. On the contrary, suppose that $F$ is not R-regular relative $domF$ at $({p^{0}},{x^{0}})$. Then there exist sequences ${p^{k}}\to{p^{0}},{p^{k}}\in domF,$ and ${v^{k}}\to{x^{0}},{v^{k}}\notin F({p^{k}})$, such that for all $k=1,2,\dots$

[TABLE]

Take any ${x^{k}}\in\Pi_{F({p^{k}})}({v^{k}})$. Due to (b) there exists a vector ${\lambda^{k}}$ such that $\sum\limits_{i=1}^{n}{\left|{\lambda_{i}^{k}}\right|}\leq M$ and

[TABLE]

It follows from the lower semicontinuity of $F$ at $(p^{0},x^{0})$ that there exists a sequence ${q^{k}}\in F({p^{k}})$ such that ${q^{k}}\to{x^{0}}$. Then $\|{{v^{k}}-{x^{k}}}\|\leq\|{{v^{k}}-{q^{k}}}\|$ and, therefore, ${x^{k}}\to{x^{0}}$.

In virtue of the boundedness of the sequence ${\lambda^{k}}$ and of the condition $\lambda_{i}^{k}{h_{i}}({p^{k}},{x^{k}})=0$, for $i\in I$, by (3.4), for all sufficiently large $k$ we obtain

[TABLE]

The latter inequality implies

[TABLE]

which contradicts (3.3). Thus $(b)\Rightarrow(a)$.

∎

Remark 3.2**.**

Theorem 3.1 is a generalization of Theorem 2 [25]. Unlike Theorem 2 [25] it does not require Lipschitz continuity of gradients ${\nabla_{x}}{h_{i}}(p,x)$ for $i=1,\dots,n$. Theorem 3.1 considers also more general notion of R-regularity relative $domF$.

Remark 3.3**.**

As follows from the proof of Theorem 2 [25] the implication $(a)\Rightarrow(b)$ holds without the assumptions of lower semicontinuity of $F$ at $({p^{0}},{x^{0}})$.

Remark 3.4**.**

Let us note that Lemma 3 of [25] is a consequence of Theorem 3.1 and Theorem (4.1) below, which says that RCRCQ for $F$ at $(p^{0},x^{0})$ and lower Lipschitz continuiuty of $F$ at $(p^{0},x^{0})$ implies $R$-regularity of $F$ at $(p^{0},x^{0})$.

In [3] we discussed in details the proof of Lemma 3 of [25] for some special functions $h_{i}$, $i=1,\dots,n$. The proof of Theorem 3.1 together with Theorem 4.1 fill some gaps in the proof of Lemma 3 of [25].

The example below shows that the assertion of Theorem 3.1 may not hold if $F$ is not lsc at a point $({p^{0}},{x^{0}})$.

Example 3.5**.**

Let $F(p)=\{x\in{\mathbb{R}^{2}}\mid{\|{{x_{1}}}\|\leq{{1,}}\ \|{{x_{2}}}\|\leq{{1,}}\ {x_{2}}-p{x_{1}}+|p|+1\leq 0\}}$, $p\in\mathbb{R}$. Then $F(p)=\{(1,-1)\}$ for all $p>0$, $F(p)=\{(-1,-1)\}$ for $p<0$ and $F(p)=\{x\in\mathbb{R}^{2}\mid{-1\leq{x_{1}}}\leq{1,\ }{x_{2}}=-1\}$ for $p=0$. Consider the point $({p^{0}},{x^{0}})$, where ${p^{0}}=0$, ${x^{0}}=(0,-1)$.

Evidently, $F$ is not lsc at $({p^{0}},{x^{0}})$. Let us take $p=\varepsilon>0$, $v=(\varepsilon,-1)$, where $\varepsilon\to 0$. It is easy to see that for given $v$ and $p$ the R-regularity condition does not hold if $\varepsilon$ is sufficiently small.

The following technical observation will be used in the sequel.

Proposition 3.6**.**

Let $p^{0}\in\mathcal{D}$. Assume that RCRCQ holds for the set-valued mapping $F$ given by (1.1) at $(p^{0},x^{0})\in\text{gph}\,F$ and $F(p)\neq\emptyset$ for $p\in V_{0}(p^{0})$. Then there exist neighbourhoods $V(p^{0})$, $V(x^{0})$ and an index set $I_{0}^{\prime}\subset I_{0}$, $|I_{0}^{\prime}|=\text{rank}\,\{\nabla_{x}h_{i}(p_{0},x_{0})\ i\in I_{0}\}$ such that for all $(p,x)\in V(p^{0})\times V(x^{0})$ vectors $\nabla_{x}h_{i}(p,x)$, $i\in I_{0}^{\prime}$ are linearly independent.

Proof.

The assertion is valid if $\nabla_{x}h_{i}(p^{0},x^{0})$, $i\in I_{0}$ are linearly independent. Suppose that $\nabla_{x}h_{i}(p_{0},x_{0})$, $i\in I_{0}$ are linearly dependent. By RCRCQ there exist neighbourhoods $V_{0}(p^{0})$, $V_{0}(x^{0})$ such that

[TABLE]

Let $\text{rank}\,\{\nabla_{x}h_{i}(p^{0},x^{0}),\ i\in I_{0}\}=k$. Then there exists indices $i_{1},\dots,i_{k}\subset I_{0}$, $i_{j}\neq i_{k}$ for $j\neq k$ such that $\nabla_{x}h_{i_{1}}(p^{0},x^{0}),\dots,\nabla_{x}h_{i_{k}}(p^{0},x^{0})$ are linearly independent. Denote $I_{0}^{\prime}=\{i_{1},\dots,i_{k}\}$. Then, by the continuity of gradients of $h_{i}$, $i=1,\dots,n$ with respect to variable $x$, $\nabla_{x}h_{i}(\cdot,\cdot)$, $i\in I_{0}^{\prime}$ are linearly independent in some neighbourhood of $(p^{0},x^{0})$. ∎

In view of Proposition 3.6, RCRCQ implies that there exists a subset $I_{0}^{\prime}\subset I_{0}$ of indices of parametric system defined by the set-valued mapping $F$ such that

[TABLE]

for $(p,x)$ in some neighbourhood of $(p^{0},x^{0})$.

4. Relaxed constant rank condition and R-regularity

It is known [5, 22] that the Mangasarian-Fromovitz constraint qualification (MFCQ) [23] for the set $F({p^{0}})$ at a point ${x^{0}}\in F({p^{0}})$ implies R-regularity of the set-valued mapping $F$ at $({p^{0}},{x^{0}})\in grF$.

We show that RCRCQ implies R-regularity of the set-valued mapping $F$.

Theorem 4.1**.**

*Assume that 1) $F$ is lsc at $({p^{0}},{x^{0}})\in grF$ relative to $domF$;

2. $F$satisfies RCRCQ at $({p^{0}},{x^{0}})\in grF$ relative to $domF\times\mathcal{H}$.*

Then $F$ is R-regular at $(p{,^{0}}{x^{0}})$ relative to $domF$.

Proof.

By Theorem 3.1, R-regularity of the mapping $F$ at $({p^{0}},{x^{0}})\in grF$ is equivalent to the fact that there exists a number $M>0$ such that for any sequences ${p_{k}}\to{p^{0}},{p_{k}}\in domF,$ ${v_{k}}\to{x^{0}},{v_{k}}\notin F({p_{k}})$, the inequality $\Lambda_{v}^{M}({p_{k}},{x_{k}})\neq\emptyset$ holds for all ${x_{k}}=x({p_{k}},{v_{k}})\in\Pi_{F({p_{k}})}({v_{k}})$ and $k$ sufficiently large.

On the contrary, suppose that there exist sequences $p_{k}\rightarrow p^{0}$, $v_{k}\rightarrow x^{0}$, $x_{k}\in F({p_{k}})$ such that $v_{k}\notin F(p_{k})$, $x_{k}\in\Pi_{F({p_{k}})}({v_{k}})$ and

[TABLE]

Due to the fact that $x^{0}\in\liminf\limits_{p\rightarrow p^{0}}F(p),$ without loss of generality, we can assume that $F(p_{k})\neq\emptyset$ for each $p_{k}$, $k=1,\dots$, and for any $\hat{x}_{k}\in\Pi_{F(p_{k})}(v_{k})$ we have $\hat{x}_{k}\rightarrow x^{0}$. In consequence, $x_{k}\rightarrow x_{0}$.

As already noted, if RCRCQ holds at $(p^{0},x^{0})$, then RCRCQ holds also at all points close to $(p^{0},x^{0})$. Without loss of generality, we can assume that RCRCQ holds at all $(p_{k},x_{k})$, $k=1,2,\dots$ . Consequently, by Lemma 3.1, $\Lambda_{v_{k}}(p_{k},x_{k})\neq\emptyset$ for all $k=1,2,\dots$.

Without loss of generality, we can assume that $(p_{k},v_{k})\in V(p^{0},x^{0})$, where by RCRCQ, $V(p^{0},x^{0})$ is such that for any $J$, $I_{0}\subset J\subset I_{0}\cup I(p_{0},x_{0})$

[TABLE]

By Lemma 2.1, $\Gamma(F(p^{k}),x^{k})=T(F(p^{k}),x^{k})$ and by the necessary optimality conditions for problem (3.2)111In the literature it is often assumed that Robinson constraint qualification holds (see for example [4]). However, it is enough to assume that the $T(F(p^{k}),x^{k})$ coincides with feasible set to the linearized problem to (3.2) (see discussion after Lemma 3.7 of [4]). we have

[TABLE]

where $\hat{\lambda}_{i}^{k}\in\mathbb{R}$, $i\in I_{0}$ and $\hat{\lambda}_{i}^{k}\geq 0$, $i\in I(p_{k},x_{k})$, $k=1,\dots$. Recall that $I(p,x):=\{i\in I\mid h_{i}(p,x)=0\}$ and $\hat{\lambda}_{i}^{k}$, $i\in I\cup I(p_{k},x_{k})$, are related to the set $\Lambda_{v_{k}}(p_{k},x_{k})$ via the relationship

[TABLE]

for some $\lambda^{k}\in\Lambda_{v_{k}}(p_{k},x_{k})$.

By Proposition 3.6, there exist neighbourhoods $V_{1}(p^{0})$, $V_{1}(x^{0})$ and indices $I_{0}^{\prime}\subset I_{0}$, $|I_{0}^{\prime}|=\text{rank}\,\{\nabla_{x}h_{i}(p_{0},x_{0})\ i\in I_{0}^{\prime}\}$ such that for all $(p,x)\in V_{1}(p^{0})\times V_{1}(x^{0})$ vectors $\nabla_{x}h_{i}(p,x)$, $i\in I_{0}^{\prime}$ are linearly independent. Hence, without loss of generality we can assume that $(p_{k},x_{k})\in V_{1}(p^{0})\times V_{1}(x^{0})$, and for every $k=1,\dots$, we have

[TABLE]

for some $\check{\lambda}_{i}^{k}\in\mathbb{R}$, $i\in I_{0}^{\prime}$.

Hence we can rewrite (4.3) as follows

[TABLE]

where $\nabla_{x}h_{i}(p_{k},x_{k})$, $i\in I_{0}^{\prime}$, are linearly independent.

Passing to a subsequence, if necessary, we may assume that for all $k\in\mathbb{N}$, $I(p_{k},x_{k})$ is a fixed set, i.e., $I(p_{k},x_{k})=I^{0}$.

By [3, Lemma 2] we have that for any $k=1,\dots$ there exists $I^{k}\subset I^{0}$ such that

[TABLE]

where $\tilde{\lambda}_{i}^{k}\geq 0$, $i\in I^{k}$ and $\nabla_{x}h_{i}(p_{k},x_{k})$, $i\in I_{0}^{\prime}\cup I^{k}$ are linearly independent. Hence we can rewrite (4.4) as

[TABLE]

Again, passing to a subsequence, if necessary, we may assume that $I^{k}=I^{\prime}$ is a fixed set.

Put $\lambda_{i}^{k}=\tilde{\lambda}_{i}^{k}\|v_{k}-p_{k}\|^{-1}$, $i\in I_{0}^{\prime}\cup I^{\prime}$, $k=1,\dots$ and $\lambda_{i}^{k}=0$, $i\in I_{0}\cup I\setminus(I_{0}^{\prime}\cup I^{\prime})$. Let us denote $\lambda^{k}=[\lambda_{i}^{k}]_{i\in I_{0}^{\prime}\cup I^{\prime}}$. We have $\lambda^{k}\in\Lambda_{v_{k}}(p_{k},x_{k})$ and, by (4.1), $\|\lambda^{k}\|\rightarrow+\infty$. Without loss of generality we may assume that $\lambda^{k}\|\lambda^{k}\|^{-1}\rightarrow\bar{\lambda}$. From (4.5) we have

[TABLE]

By passing to the limit in (4.6) we obtain

[TABLE]

where $\|\bar{\lambda}\|=1$. This contradicts the fact that for $k=1,\dots$

[TABLE]

i.e. vectors $\nabla_{x}h_{i}(p_{0},x_{0}),\ i\in I_{0}^{\prime}\cup I^{\prime}$ are linearly independent. ∎

5. Lipschitz-likeness of $F$

The following theorem provides the relationships between R-regularity and pseudo-Lipschitzness of the set-valued mapping $F$ defined on a normed space $\mathcal{G}$.

Theorem 5.1**.**

Assume that $F$ is R-regular at a point $({p^{0}},{x^{0}})\in grF$ relative to $domF$. Then $F$ is pseudo-Lipschitzian at this point relative to $domF$.

Proof.

If $F$ is R-regular at a point $({p^{0}},{x^{0}})\in grF$ relative to $domF$, this means that there are numbers $M>0$,$\delta>0$, $\varepsilon>0$ such that

[TABLE]

for all $x\in{V_{\varepsilon}}({x^{0}})$ and all $p\in{V_{\delta}}({p^{0}})\cap domF$.

Denote $l=\max\{{l_{i}}\left|{i=1,\dots,p\}}\right.$ where ${l_{i}}$ are Lipschitz constants for functions ${h_{i}}(p,x)$ on some set $V({p^{0}})\times V({x^{0}})$. Choose numbers $\delta$ and $\varepsilon$ such that ${V_{\delta}}({p^{0}})\subset V({p^{0}}),{V_{\varepsilon}}({x^{0}})\subset V({x^{0}})$. Then

[TABLE]

for all $p\in{V_{\delta}}({p^{0}})\cap domF$.

This means that $F$ is Lipschitz lower continuous at $({p^{0}},{x^{0}})$ relative to $domF$ and, consequently (see (2.1)), $F(p)\cap{V_{\varepsilon}}({x^{0}})\neq\emptyset$ for all $p\in{V_{\eta}}({p^{0}})\cap domF$, where $\eta<\min\{\delta,\varepsilon/l\}$. Let $p,\tilde{p}\in{V_{\eta}}({p^{0}})\cap domF$ and let $\tilde{x}\in F(\tilde{p})\cap{V_{\varepsilon}}({x^{0}})$. Then, from (5.1) follows

[TABLE]

The last inequality is equivalent to

[TABLE]

∎

Remark 5.2**.**

Let us note that in Theorem 5.1 we only need to assume that all the functions $f$ and $h_{i}$, $i=1,\dots,n$ are locally Lipschitz continuous (may not be differentiable) near $({p^{0}},{x^{0}})\in grF$.

By Theorem 4.1 we obtain the following result

Theorem 5.3**.**

*Assume that 1) $F$ is lsc at $({p^{0}},{x^{0}})\in grF$ relative to $domF$;

2. $F$satisfies RCRCQ at $({p^{0}},{x^{0}})\in grF$ relative to $domF\times\mathcal{H}$.*

Then $F$ is pseudo-Lipschitzian at this point relative to $domF$.

Let us note that for some particular functions $h_{i}(\cdot,\cdot)$, $i\in I\cup I_{0}$ Theorem 5.3 has been already proved in [3].

In the finite-dimensional setting, when both $\mathcal{H}$ and $\mathcal{G}$ are finite-dimensional spaces the results analogous to Theorem 5.3 can be obtained via properties of the optimal value function defined as $\varphi(p):=\inf\{f(p,x)\mid x\in F(p)\}$ and the solution set $S(p):=\{x\in F(p)\mid{f(p,x)\leq\varphi(p)}\}$.

Definition 5.4**.**

Let ${p^{0}}\in domF$. A mapping $F$ is locally bounded at ${p^{0}}$ if there exist a neighborhood $V({p^{0}})$ and a bounded set ${Y_{0}}$ such that $F(p)\subset{Y_{0}}$ for all $p\in V({p^{0}})$.

Theorem 5.5**.**

*Assume that 1) F is locally bounded at ${p^{0}}$and functions $f,{h_{i}}$ are Lipschitz continuous on a set $V({p^{0}})\times({Y_{0}}+\varepsilon B)$, where $\varepsilon>2diam{Y_{0}}$;

2. $F$is R-regular at $({p^{0}},{x^{0}})\in grF$ relative to $dom\,F$.

Then $\varphi$ is Lipschitz continuous on some set $V({p^{0}})\cap domF$.*

Proof.

Let ${l_{0}}>0$ and ${l_{i}}>0$ be Lipschitz constants for $f$ and ${h_{i}}$ on a set $V({p^{0}})\times({Y_{0}}+\varepsilon B)$. Then due to Lemma 3.68 [22]

[TABLE]

Let us take any $p,\tilde{p}\in V({p^{0}})$ and $\tilde{x}\in S(\tilde{p})$. Then without loss of generality

[TABLE]

Similarly, one can obtain $\varphi(\tilde{p})-\varphi(p)\leq({l_{0}}+\alpha M\max\{{l_{i}}:i\in I\cup{I_{0}}\})\|{p-\tilde{p}}\|.$ ∎

6. Application to bilevel programming

Consider a bilevel programming problem (BLPP):

[TABLE]

where $F(p)$ is defined in (1.1), $J=\{1,\dots,l\}$, functions $G(p,x)$, ${g_{j}}(p)$, $f(p,x)$ and ${h_{i}}(p,x)$ are continuously differentiable (see e.g. monograph [7]).

The point $(p,x)$ is said to be a feasible point to the problem (6.4) if $p\in P$, $x\in S(p)$. A feasible $({p^{0}},{x^{0}})$ is called a solution (local solution) of the problem (6.4) if $G({p^{0}},{x^{0}})\leq G(p,x)$ for all feasible points $(p,x)$ (for all feasible points from some neighborhood of $({p^{0}},{x^{0}})$).

The problem (6.4) can be equivalently reformulated as the following one-level problem

[TABLE]

where $\varphi(p)=\inf\{f(p,x)|x\in F(p)\}$ is the optimal value function of the lower-level problem. Main difficulty in solving problem (6.5) comes from the nonsmoothness of the value function $\varphi(p)$. Ye and Zhu [27] introduced the concept of partial calmness which allowed to move the nonsmooth constraint from the feasible set to the objective function.

Let $({p^{0}},{x^{0}})$ be a feasible point of the problem (6.4). The problem (6.4) in the form (6.5) is called partially calm at $({p^{0}},{x^{0}})$, if there exist a number $\mu>0$ and a neighborhood $V$ of the point $({p^{0}},{x^{0}},0)$ in ${\mathbb{R}^{n}}\times{\mathbb{R}^{m}}\times\mathbb{R}$ such that $G(p,x)-G({p^{0}},{x^{0}})+\mu\left|u\right|\geq 0$ for all $(p,x,u)\in V$ such that $p\in\mathcal{G}$, $x\in S(p)$, $f(p,x)-\varphi(p)+u=0$.

In [27] it was proved that the problem (6.4) in the form (6.5) is partial calm at its local solution $({p^{0}},{x^{0}})$if and only if there exists a number $\mu>0$ such that $({p^{0}},{x^{0}})$ is a local solution of the partially penalized problem

[TABLE]

Ye and Zhu [27] showed that the problem (6.4) with a linear in $(p,x)$ lower-level problem is partially calm.

Let us define the set-valued mapping $S:p\rightrightarrows S(p)$ and consider the bilevel program (6.4) under the following assumption:

1. (H1)

$P\cap domS=P\cap domF$.

Denote ${h_{0}}(p,x):=f(p,x)-\varphi(p)$. Introduce the sets

[TABLE]

and $C(p):=\{x\in\mathbb{R}^{m}\mid(p,x)\in C\}.$

Theorem 6.1**.**

Let $({p^{0}},{x^{0}})$ be a local solution of problem (6.4). Suppose that the mapping $S$ is R-regular at $({p^{0}},{x^{0}})$ and the function $G$ is Lipschitz continuous on the set $D$ with Lipschitz constant ${l_{0}}>0$. Then there exists a number ${\mu_{0}}>0$ such that for any $\mu>{\mu_{0}}$ the point $({p^{0}},{x^{0}})$is a local solution to the problem

[TABLE]

Proof.

It follows from (H1) that $d(x,C(p))=d((p,x),\{p\}\times C(p))\geq d((p,x),C)$ for all $(p,x)\in D$. In virtue of Proposition 2.4.3 of [6], for all $\alpha>{l_{0}}$, the point $({p^{0}},{x^{0}})$ is a solution of the problem

[TABLE]

Then $d((p,x),C)\leq d(x,C(p))$ for all $(p,x)\in D$ and, therefore, $G(p,x)+\alpha d(x,C(p))\geq G({p^{0}},{x^{0}})+\alpha d({x^{0}},C({p^{0}}))=G({p^{0}},{x^{0}})$.

Since R-regularity for $S$ implies R-regularity for the set-valued $p\rightrightarrows C(p)$, there exists a neighbourhood $V(p{{}^{0}},{x^{0}})$ such that $G(p,x)+\alpha M\max\{0,{h_{0}}(p,x)\}\geq G({p^{0}},{x^{0}})$ for all $(p,x)\in D\cap V(p{{}^{0}},{x^{0}})$. The last inequality is equivalent to the assertion of the theorem with ${\mu_{0}}={l_{0}}M$. ∎