Relaxed constant positive linear dependence constraint qualification and its application to bilevel programs

A constraint qualification is a condition imposed on the constraint region of a mathematical program under which the Karush-Kuhn-Tucker (KKT) condition holds at any local optimal solution. Other than guaranteeing KKT condition holding at all local optimal solutions, some constraint qualifications also lead to existence of error bounds to the feasible region and hence play a key role in convergence analysis of certain computational methods. Hence studying constraint qualifications is essential in both theoretical and numerical points of view.

For smooth nonlinear programs with equality and inequality constraints, the classical constraint qualifications are the linear independence constraint qualification (LICQ), Mangasarian-Fromovitz constraint qualification (MFCQ), and the linear constraint qualification (LCQ), i.e., all functions in the equality and inequality constraints are affine. These three classical constraint qualifications may be too restrictive for many problems. Janin [16] relaxed LICQ and proposed the constant rank constraint qualification (CRCQ), which neither implies nor is implied by MFCQ. The concept of CRCQ was weakened to the relaxed constant rank constraint qualification (RCRCQ) which is shown to be a constraint qualification by Minchenko and Stakhovshi [19]. Qi and Wei [26] introduced the concept of the constant positive linear dependence (CPLD) condition which is weaker than both CRCQ and MFCQ. CPLD was shown to be a constraint qualification by Andreani, Martínez and Schuverdt in [2].

In [1], Andreani, Haeser, Schuverdt and Silva introduced the following relaxed version of CPLD for a system of smooth equality and inequality constraints:

$$g_{i}(x)\leq 0,\ i=1,\cdots,n,\ \ h_{i}(x)=0,\ i=1,\cdots,m,$$

where $g_{i},h_{i}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ are smooth at $x^{*}$, a feasible solution. $x^{*}$ is said to satisfy the relaxed constant positive linear dependence constraint qualification $(\rm RCPLD)$ if there exists $\mathbb{U}(x^{*})$, a neighborhood of $x^{*}$ such that

• (i)

  $\{\nabla h_{i}(x)\}_{i=1}^{m}$ has the same rank for every $x\in\mathbb{U}(x^{*})$.

• (ii)

  Let $J\subseteq\{1,\cdots,m\}$ be such that $\{\nabla h_{i}(x^{*})\}_{i\in J}$ is a basis for span$\{\nabla h_{i}(x^{*})\}_{i=1}^{m}$. For every $I\subseteq I_{g}^{*}:=\{i:g_{i}(x^{*})=0\}$, if $\{\nabla g_{i}(x^{*})\}_{i\in I}\cup\{\nabla h_{i}(x^{*})\}_{i\in J}$ is positive linearly dependent, i.e., there exist scalars $\lambda_{i}\geq 0$, $i\in I$, $\mu_{i}$, $i\in J$ not all zero such that

  $\displaystyle 0=\sum_{i\in I}\lambda_{i}\nabla g_{i}(x^{*})+\sum_{i\in J}\mu_{i}\nabla h_{i}(x^{*}),$

  then $\{\nabla g_{i}(x)\}_{i\in I}\cup\{\nabla h_{i}(x)\}_{i\in J}$ is linearly dependent for every $x\in\mathbb{U}(x^{*})$.

It is easy to see that in the case where either LCQ or MFCQ holds, RCPLD also holds. Hence RCPLD is weaker than LCQ and MFCQ. In [1], the authors not only showed that RCPLD is a constraint qualification but also proved that if all functions $g_{i},h_{i}$ have second-order derivatives at all points near the point $x^{*}$, then RCPLD is a sufficient condition for the error bound property: $\exists\alpha>0,\mathbb{U}(x^{*})$, a neighborhood of $x^{*}$ such that

$$d_{{\cal F}}(x)\leq\alpha(\|g_{+}(x)\|+\|h(x)\|),\quad\forall x\in\mathbb{U}(x^{*}),$$

where ${{\cal F}}:=\{x|g(x)\leq 0,h(x)=0\}$, $g(x)=(g_{1}(x),\dots,g_{n}(x))$, $h(x)=(h_{1}(x),\dots,h_{m}(x))$, $d_{{\cal F}}(x)$ is the distance from $x$ to set ${{\cal F}}$, $\|\cdot\|$ denotes any norm and $g_{+}(x):=\max\{0,g(x)\}$, where the maximum is taken component-wise. Moreover as an open question, in [1], a question was asked on whether or not it was possible to prove the error bound property without imposing the second-order differentiability of all functions. In Guo, Zhang and Lin [14], it was shown that the error bound property holds under RCPLD without imposing the second order differentiability of all functions. Other than using it as a constraint qualification to ensure the KKT condition holds, RCPLD is also used in the convergence analysis of the augmented Lagrangian method to obtain a KKT point (see e.g., [1, 3, 15, 28]). Recently, Guo and Ye [13, Corollary 3] extended RCPLD to the case where there is an extra abstract constraint set and showed that it is still a constraint qualification. In [11, Definition 4.3], a version of RCPLD called MPEC RCPLD was introduced for the mathematical programs with equilibrium constraints (MPEC) and was shown in [10, Corollary 4.1] that it is a constraint qualification for M-stationary conditions. Moreover, it was shown in [14, Theorem 5.1] that the RCPLD for MPECs ensures the error bound property under the assumption of strict complementarity.

In this paper, we extend RCPLD to the following very general feasibility system:

$$\begin{array}[]{l}g_{i}(x)\leq 0,\ i=1,\cdots,n,\\ h_{i}(x)=0,\ i=1,\cdots,m,\\ (G(x),H(x))\in\Omega^{p},\\ x\in C,\end{array}$$

(1.1)

where $\Omega^{p}:=\{(y,z)\in\mathbb{R}^{p}\times\mathbb{R}^{p}|0\leq y\perp z\geq 0\}$ is the $p$th dimensional complementarity set, $C:=C_{1}\times C_{2}\times\cdots\times C_{l}$ with $C_{i}\subseteq\mathbb{R}^{q_{i}}$ closed, $i=1,\cdots,l$, $q_{1}+\cdots+q_{l}=d$, the functions $g_{i}:\mathbb{R}^{d}\to\mathbb{R},i=1,\cdots,n$, are locally Lipschitz continuous and $h_{i}:\mathbb{R}^{d}\to\mathbb{R},i=1,\cdots,m$, $G,H:\mathbb{R}^{d}\rightarrow\mathbb{R}^{p}$, are continuously differentiable at $x^{*}$, a feasible solution.

Denote the feasible region of system $(\ref{feasibilitys})$ by ${\cal F}$. For a feasible point $x^{*}\in{\cal F}$, we define the following index sets:

	$\displaystyle I_{g}^{*}:=\{i=1,\cdots,n:g_{i}(x^{*})=0\},$
	$\displaystyle\mathcal{I}^{*}:=\{i=1,\cdots,p:0=G_{i}(x^{*})<H_{i}(x^{*})\},$
	$\displaystyle\mathcal{J}^{*}:=\{i=1,\cdots,p:0=G_{i}(x^{*})=H_{i}(x^{*})\},$
	$\displaystyle\mathcal{K}^{*}:=\{i=1,\cdots,p:G_{i}(x^{*})>H_{i}(x^{*})=0\}.$

In this paper we show that RCPLD for the nonsmooth feasibility system is a constraint qualification for any optimization problem with a Lipschitz objective function and the constraints described as in the feasibility system (1.1). Moreover with some extra conditions, we will show that RCPLD is a sufficient condition for the following error bound property: $\exists\alpha>0,\mathbb{U}(x^{*})$, a neighborhood of $x^{*}$ such that

$$d_{{\cal F}}(x)\leq\alpha(\|g_{+}(x)\|+\|h(x)\|+\sum_{i=1}^{p}d_{\Omega}(G_{i}(x),H_{i}(x))),\quad\forall x\in\mathbb{U}(x^{*})\cap C,$$

(1.3)

where $\Omega:=\Omega^{1}=\{(y,z)\in\mathbb{R}^{2}|\ 0\leq y\perp z\geq 0\}$.

One of the motivations to extend the concept of RCPLD to the nonsmooth system (1.1) is to study the constraint qualification and optimality condition for the following bilevel program:

	$\displaystyle({\rm BP})~{}~{}~{}~{}~{}~{}\min$		$\displaystyle F(x,y)$
	$\displaystyle{\rm s.t.}$		$\displaystyle y\in S(x),\ G(x,y)\leq 0,\ H(x,y)=0,$

where $S(x)$ denotes the solution set of the lower level program

$\displaystyle({\rm P}_{x})~{}~{}~{}~{}~{}~{}~{}~{}\min_{y\in Y(x)}\ f(x,y),$

and $Y(x):=\{y\in\mathbb{R}^{s}:g(x,y)\leq 0,h(x,y)=0\}$, $F:\mathbb{R}^{d}\times\mathbb{R}^{s}\rightarrow\mathbb{R}$ is locally Lipschitz continuous, $G:\mathbb{R}^{d}\times\mathbb{R}^{s}\rightarrow\mathbb{R}^{p}$ and $H:\mathbb{R}^{d}\times\mathbb{R}^{s}\rightarrow\mathbb{R}^{q}$ are continuously differentiable, $f:\mathbb{R}^{d}\times\mathbb{R}^{s}\rightarrow\mathbb{R}$, $g:\mathbb{R}^{d}\times\mathbb{R}^{s}\rightarrow\mathbb{R}^{m},h:\mathbb{R}^{d}\times\mathbb{R}^{s}\rightarrow\mathbb{R}^{n}$ are continuously differentiable and twice continuously differentiable with respect to variable $y$.

Bilevel programs naturally fall in the domain of global optimization since in the lower level problem, the global optimal solution is always required in either optimality conditions or numerical algorithms. A popular approach to deal with (BP) is to replace the set of global solutions $S(x)$ by the KKT optimality conditions of the lower level problem. This reformulation is based on the fact that if the lower level problem $(P_{x})$ is a convex program for each fixed $x$ and certain constraint qualification holds, then $y\in S(x)$ if and only if its KKT optimality condition holds. But due to the introduction of multipliers for the lower level problem as extra variables, the resulting reformulation may not be equivalent to the original bilevel program even in the case where $(P_{x})$ is convex but the multipliers are not unique. For the discussion of this issue and the recent new results, the reader is referred to recent paper [31] and the reference within for more discussions. If the lower level problem $(P_{x})$ is not a convex program for each fixed $x$, the KKT condition for the lower level problem is usually only a necessary but not sufficient condition for optimality and moreover, as pointed out by Mirrlees in [20], such a reformulation by the KKT condition may miss out the true optimal solution of the original bilevel program.

Instead of using $y\in S(x)$ as a constraint in (BP), Outrata [25] proposed to replace it with $f(x,y)-V(x)=0,y\in Y(x)$ where $V(x):=\inf_{y\in Y(x)}f(x,y)$ is the value function of the lower level program for a numerical consideration. This so-called value function approach was further used in Ye and Zhu [32] and later in other papers such as [7, 8, 21, 22, 23, 24] to derive various necessary optimality conditions under the partial calmness condition. The partial calmness condition for the value function reformulation of the bilevel program, however, is a very strong assumption. To derive a necessary optimality condition under weaker assumptions, Ye and Zhu [33] proposed the following combined program where both the value function constraint and the KKT condition of the lower level program are used as constraints:

	$\displaystyle({\rm CP})~{}~{}~{}~{}~{}~{}\min_{x,y,u,v}$		$\displaystyle F(x,y)$
	$\displaystyle{\rm s.t.}$		$\displaystyle f(x,y)-V(x)\leq 0,\ G(x,y)\leq 0,H(x,y)=0,$
			$\displaystyle\nabla_{y}f(x,y)+\nabla_{y}g(x,y)^{T}u+\nabla_{y}h(x,y)^{T}v=0,$
			$\displaystyle(-g(x,y),u)\in\Omega^{m}.$

As discussed in [33], such a reformulation can avoid some difficulties caused by using the value function or the classical KKT approach alone. In [30], necessary optimality conditions in the form of Mordukhovich (M-) stationary condition for (CP) is studied under the partial calmness/weak calmness condition. These conditions, however, may not be easy to verify.

Note that the inclusion of the value function makes the problem nonsmooth since the value function is usually nonsmooth but under some reasonable assumptions on the lower level problem, the value function is Lipschitz continuous. Hence the feasible set of (CP) is a special case of the general feasibility system (1.1). However as it was shown in Ye and Zhu [33, Proposition 1.3], the no nonzero abnormal multiplier constraint qualification (NNAMCQ) as defined in Definition 4.1 never holds for (CP). Being able to deal with nonsmooth inequality constraints in RCPLD would allow us to present verifiable constraint qualifications and exact penalty for the reformulation of the bilevel program in which the value function is used, such as (CP).

The rest of the paper is organized as follows. In Section 2, we present basic definitions as well as some preliminaries which will be used in this paper. In Section 3, we show that RCPLD is a constraint qualification and a sufficient condition for error bound properties under certain regularity conditions. We introduce some sufficient conditions for RCPLD, which are easier to verify in Section 4. In Section 5, we apply RCPLD to the bilevel program.

We adopt the following standard notation in this paper. For any two vectors $a$ and $b$, we denote by either $\langle a,b\rangle$ or $a^{T}b$ their inner product. Given a differentiable function $G:\mathbb{R}^{d}\rightarrow\mathbb{R}$, we denote its gradient vector by $\nabla G(x)\in\mathbb{R}^{d}$. For a differentiable mapping $\Phi:\mathbb{R}^{d}\to\mathbb{R}^{n}$ with $n\geq 2$ and a vector $x\in\mathbb{R}^{d}$, we denote by $\nabla\Phi(x)\in\mathbb{R}^{n\times d}$ the Jacobian matrix of $\Phi$ at $x$. For a set $C\subseteq\mathbb{R}^{d}$, we denote by int $C$, co $C$, $\bar{C}$, bd $C$ and $d_{C}(x)$ the interior, the convex hull, the closure, the boundary of $C$ and the distance from $x$ to $C$, respectively. We denote by $|{\cal I}|$ the cardinality of index set ${\cal I}\subseteq\{1,\dots,n\}$. For any vector $v\in\mathbb{R}^{n}$ and a given index set ${\cal I}\subseteq\{1,\dots,n\}$, we use $v_{\cal I}$ to denote the vector in $\mathbb{R}^{|{\cal I}|}$ with components $v_{i},i\in{\cal I}$. For a matrix $A\in\mathbb{R}^{n\times m}$, $A^{T}$ denotes its transpose, $r(A)$ denotes its rank. We denote by ${\mathbb{B}}_{\delta}(\bar{x})$ the open ball center at $\bar{x}$ with radius $\delta>0$ and by ${\mathbb{B}}$ the open unit ball center at the origin. Unless otherwise specified we denote by $\|\cdot\|$ any norm in the finite dimensional space.

In this section, we present some background materials on variational analysis which will be used throughout the paper. Detailed discussions on these subjects can be found in [4, 5, 21, 27].

Given a set $C\subseteq\mathbb{R}^{d}$, its Fréchet/regular normal cone at $z\in C$ is defined by

$$\mathaccent 866{\mathcal{N}}_{C}(z):=\{v\in\mathbb{R}^{d}:\langle v,z^{\prime}-z\rangle\leq o\big{(}\|z-z^{\prime}\|\big{)}\;\mbox{ for all }\;z^{\prime}\in C\}$$

and the limiting/Mordukhovich/basic normal cone to $C$ at point $z$ is defined by

$$\mathcal{N}_{C}(z)=\{\lim_{k\rightarrow\infty}v_{k}:v_{k}\in\mathaccent 866{\mathcal{N}}_{C}(z_{k}),\ \ z_{k}\in C,z_{k}\rightarrow z\}.$$

A set $C\subseteq\mathbb{R}^{d}$ is Clarke regular at $z$ if it is locally closed at $z$ and $\mathcal{N}_{C}(z)=\mathaccent 866{\mathcal{N}}_{C}(z)$ [27, Definition 6.4]. Note that for $C:=C_{1}\times C_{2}\times\cdots\times C_{l}$ with $C_{i}$ closed, $i=1,\cdots,l$, by [27, Proposition 6.41], $C$ is regular at $z$ if and only if each $C_{i}$ is regular at $z_{i}$.

The exact expressions for the limiting normal cone of the complementarity set present below will be useful. In this paper we denote by $\Omega^{p}$ the complementarity set and when $p=1$, $\Omega:=\Omega^{1}=\{(y,z)\in\mathbb{R}^{2}|\ 0\leq y\perp z\geq 0\}$.

Let $f:\mathbb{R}^{d}\rightarrow\overline{\mathbb{R}}$ be a lower semicontinuous function and finite at $x\in\mathbb{R}^{d}$. We define the Fréchet/regular subdifferential ([27, Definition 8.3]) of $f$ at $x$ as

$\displaystyle\mathaccent 866{\partial}f(x):=\{\zeta\in\mathbb{R}^{d}:f(x^{\prime})-f(x)-\langle\zeta,x^{\prime}-x\rangle\geq o\|x^{\prime}-x\|\},$

and the limiting/Mordukhovich/basic subdifferential of $f$ at ${x}$ as

$\displaystyle\partial f({x}):=\{\lim_{k\rightarrow\infty}\xi_{k}:\xi_{k}\in\mathaccent 866{\partial}f(x_{k}),x_{k}\rightarrow x,f(x_{k})\rightarrow f(x)\}.$

Let $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ be Lipschitz continuous at $x$. We say that $f$ is subdifferentially/Clarke regular at $x$ provided that $\partial f({x})=\mathaccent 866{\partial}f(x)$ [27, Corollary 8.11].

The following proposition collects some useful properties and calculus rules of the limiting subdifferential.

Taking into account that all norms in a finite dimensional space are equivalent, the following formula for distance function to the complementarity set $\Omega$ can be used in the error bound property (1.3).

In Theorem 3.1, we need to calculate $\partial\phi_{0}(x)$, where

$\displaystyle\phi_{0}(x):=\sum_{i=1}^{n}\max\{0,g_{i}(x)\}+\sum_{i=1}^{m}|h_{i}(x)|+\sum_{i=1}^{p}|\min\{G_{i}(x),H_{i}(x)\}|.$

In order to calculate $\partial\phi_{0}(x)$, we define the following index sets for each $x$:

	$\displaystyle A(x):=\{i=1,\cdots,n:g_{i}(x)\geq 0\},$
	$\displaystyle\mathcal{I}(x):=\{i=1,\cdots,p:G_{i}(x)<H_{i}(x)\},$
	$\displaystyle\mathcal{J}(x):=\{i=1,\cdots,p:G_{i}(x)=H_{i}(x)\},$		(2.2)
	$\displaystyle\mathcal{K}(x):=\{i=1,\cdots,p:G_{i}(x)>H_{i}(x)\}.$

From the calculus rules in Proposition 2.2, we have the following estimate for the limiting subdifferential of $\phi_{0}$.

Proof. For $i=1,\cdots,p$, let $F_{i}(x):=|f_{i}(x)|$ and $f_{i}(x):=\min\{G_{i}(x),H_{i}(x)\}$. From the chain rule in Proposition 2.2(ii), we have $\partial F_{i}(x^{*})\subseteq\{\partial(\mu f_{i})(x^{*}):\mu\in\partial|\cdot|(f_{i}(x^{*}))\}.$ We divide the analysis into three cases.

• Case 1:

  $i\in\mathcal{I}(x^{*})$. In this case we have $\ G_{i}(x^{*})<H_{i}(x^{*})$ and hence $f_{i}(x^{*})=G_{i}(x^{*})$. By the chain rule we have $\partial F_{i}(x)=\{\nabla G_{i}(x^{*})\}$ if $G_{i}(x^{*})>0$ and $\partial F_{i}(x)=\{-\nabla G_{i}(x^{*})\}$ if $G_{i}(x^{*})<0$. If $G_{i}(x^{*})=0$, then $\partial|\cdot|(f_{i}(x^{*}))=[-1,1]$ and hence $\partial F_{i}(x)\subseteq\{\mu\nabla G_{i}(x^{*}):\mu\in[-1,1]\}$.

• Case 2:

  $i\in\mathcal{K}(x^{*})$. Similarly as in Case 1, we can show $\partial F_{i}(x)\subseteq\{\mu\nabla H_{i}(x^{*}):\mu\in[-1,1]\}$.

• Case 3:

  $i\in\mathcal{J}(x^{*})$. In this case, $f_{i}(x^{*})=G_{i}(x^{*})=H_{i}(x^{*})$. If $G_{i}(x^{*})=H_{i}(x^{*})<0$, then $\partial|\cdot|(f_{i}(x^{*}))=\{-1\}$ and $\partial F_{i}(x^{*})\subseteq\partial(-f_{i})(x^{*})$. Since $-f_{i}(x)=\max\{-G_{i}(x),-H_{i}(x)\}$, by Proposition 2.2(iii), $\partial F_{i}(x)=co\{-\nabla G_{i}(x^{*}),-\nabla H_{i}(x^{*})\}$. If $G_{i}(x^{*})=H_{i}(x^{*})>0$, then $\partial|\cdot|(f_{i}(x^{*}))=\{1\}$ and $\partial F_{i}(x^{*})\subseteq\partial f_{i}(x^{*})$. It follows by Proposition 2.2(iii) that $\partial F_{i}(x)\subseteq\{\nabla G_{i}(x^{*}),\nabla H_{i}(x^{*})\}$. If $G_{i}(x^{*})=H_{i}(x^{*})=0$, then $\partial|\cdot|(f_{i}(x^{*}))=[-1,1]$, $\partial F_{i}(x)\subseteq\{\partial(\mu f_{i})(x^{*}):\mu\in[-1,1]\}$. If $\mu>0$, then $\partial(\mu f_{i})(x^{*})\subseteq\mu\partial f_{i}(x^{*})\subseteq\mu\{\nabla G_{i}(x^{*}),\nabla H_{i}(x^{*})\}$. If $\mu<0$, then $\partial(\mu f_{i})(x^{*})=-\mu\partial(-f_{i})(x^{*})=-\mu co\{-\nabla G_{i}(x^{*}),-\nabla H_{i}(x^{*})\}$. Hence

  $$\partial F_{i}(x^{*})\subseteq\{\lambda v:\lambda\geq 0,v\in co\{-\nabla G_{i}(x^{*}),-\nabla H_{i}(x^{*})\}\cup\{\nabla G_{i}(x^{*}),\nabla H_{i}(x^{*})\}\}.$$

Summarizing the above cases, we have that for any $x^{*}$,

$$\partial|\min\{G_{i}(x^{*}),H_{i}(x^{*})\}|\subseteq\left(\begin{array}[]{l}\nabla G(x^{*})\\ \nabla H(x^{*})\end{array}\right)^{T}(-\lambda^{G},-\lambda^{H}),$$

where $(\ref{multipliers})$ holds.

Since $g_{i}(x),i=1,\cdots,n$, $|h_{i}(x)|,j=1,\cdots,m$ and $|\min\{G_{i}(x),H_{i}(x)\}|,i=1\cdots,p$ are all Lipschitz continuous around $x^{*}$, by the calculus rules in Proposition 2.2 and (2.3), there exist parameters $\lambda_{i}^{g}\geq 0,i\in A(x^{*})$, $\lambda_{i}^{h},i=1,\cdots,m$ and $\lambda_{i}^{G},\lambda_{i}^{H}$, $i=1,\cdots,p$ satisfying (2.3) such that

$\displaystyle\partial\phi_{0}(x^{*})\subseteq\sum_{i\in A(x^{*})}\lambda_{i}^{g}\partial g_{i}(x^{*})+\sum_{i=1}^{m}\lambda_{i}^{h}\nabla h_{i}(x^{*})-\sum_{i=1}^{p}\lambda_{i}^{G}\nabla G_{i}(x^{*})-\sum_{i=1}^{p}\lambda_{i}^{H}\nabla H_{i}(x^{*}).$