1 Introduction

The so-called feasibility problem asks for a point contained in the intersection of a finite collection of constraint sets. More precisely, given a finite family of sets (called geometric constraints) \(F_{1}, F_{2},\ldots , F_{r}\) (r is a natural number) in the Euclidean space \({\mathbb {R}}^{n},\) the corresponding feasibility problem takes the form

$$\begin{aligned} \text{ Find } \quad x \in F:= \bigcap _{j=1}^{r} F_{j}. \end{aligned}$$
(1.1)

A feasibility problem is said to be consistent when it has a solution, i.e., whenever F is a nonempty set, otherwise is said to be inconsistent. This seemingly simple problem provides a modeling framework with great flexibility and power. In many situations of practical interest, it is difficult to find such a point in F directly.

In this study, we consider the important feasibility problem of projecting onto the intersection of the geometric constraints \(F_{1}, F_{2},\ldots , F_{r},\) frequently also referred to as the constrained best approximation problem. The constrained best approximation problem of an arbitrary point \(x \in {\mathbb {R}}^{n}\) in the intersection of geometric constraints (i.e., F) is of central significance in diverse areas of mathematics and engineering. Many authors have researched on this problem and most of them naturally involve the concept of differentiability and variants of subdifferentials with respect to the constraint functions. We refer the reader to [5,6,7, 9, 10] and the references therein for more review and discussion on projecting methods and applications.

Many problems in optimization and approximation theory can be expressed into one of the following two types: one is a system of inequalities:

$$\begin{aligned} g_{j}(x)\le 0, \ \ j=1,2,\ldots ,r, \end{aligned}$$
(1.2)

and the other is a minimization problem:

$$\begin{aligned} \left\{ \begin{array}{l} minimize \ f(x), \\ \text{ subject } \text{ to } \ \ g_{j}(x)\le 0, \ \ j=1,2,\ldots ,r, \end{array} \right. \end{aligned}$$
(1.3)

where \(f, g_{j}:{\mathbb {R}}^{n} \longrightarrow {\mathbb {R}}\) \((j=1,2,\ldots ,r)\) are real valued functions. In the above problems, necessary and sufficient conditions for optimal solutions play an important role. The set of feasible solutions of the systems (1.2) and (1.3) is of the form

$$\begin{aligned} F:= \bigcap _{j=1}^{r} \{x \in {\mathbb {R}}^n: g_j(x) \le 0\}, \end{aligned}$$
(1.4)

where it is assumed that F is a nonempty closed set. Also, \( f: {\mathbb {R}}^{n} \longrightarrow {\mathbb {R}}\) is called the objective function.

In the sequel, we consider the following minimization problem:

$$\begin{aligned} \left\{ \begin{array}{l} minimize \ f(x), \\ \text{ subject } \text{ to } \ x \in F, \end{array} \right. \end{aligned}$$
(1.5)

in which the set \(F:= \bigcap _{j=1}^{r} F_j\) is defined by (1.1), where \(F_{1}, F_{2},\ldots , F_{r}\) are called the geometric constraints. Throughout the paper, we assume that F is a nonempty closed set.

The main goal of this paper is (for the first time) to provide necessary and sufficient conditions for the constrained best approximation problem of an arbitrary point \(x \in {\mathbb {R}}^n\) in F with geometric constraints, where F is defined by (1.1). In this case, we assume that the geometric constraints \(F_{1}, F_{2},\ldots , F_{r}\) are pseudoconvex with respect to the classical contingent cone of \(F_{1}, F_{2},\ldots , F_{r},\) respectively, at a reference feasible point \(x \in F.\) Indeed, we show under mild conditions on the geometric constraints \(F_{1}, F_{2},\ldots , F_{r}\) that the “perturbation property” of the constrained best approximation from a nonempty closed set \(K \cap F\) is characterized by the “convex conical hull intersection property” (CCHIP in short) at a reference feasible point \(x \in F,\) where F is defined by (1.1) and K is a nonempty closed convex set in \({\mathbb {R}}^n\) such that \(K \cap F \ne \emptyset .\) We do this by first establishing a dual cone characterization of the contingent cone of the set F. Finally, we present the “Lagrange multiplier characterizations” of the constrained best approximation. Our results extend and completely solve the constrained best approximation problem, in a more general case, and not only recapture the corresponding known results of [2, 3, 5, 9,10,11,12, 15, 16] and the references therein, but (in a particular case) also allow applications to problems such that their constraint functions \(g_j, \ j = 1, 2, \ldots ,r,\) are quasiconvex as they guarantee the convexity of the sublevel sets (in particular, the geometric constraints). Simple numerical examples illustrate the nature of our results.

The presentation of the paper is as follows. Some definitions, basic facts and important auxiliary results are presented in Sect. 2. In Sect. 3, we give a dual cone characterization of the normal cone of the set F. In Sect. 4, we first present a dual cone characterization of the contingent cone of the set F at a reference feasible point \(x \in F.\) Next, by using this fact, we give characterizations of the “perturbation property” of the constrained best approximation. Finally, we obtain the “Lagrange multiplier characterizations” of the constrained best approximation. Several examples are given to illustrate and clarify our results. The conclusions and applications are presented in Sect. 5.

2 Preliminaries

Let \(\langle x, u \rangle \) denote the inner product of two vectors x and u in the n-dimensional Euclidean space \({\mathbb {R}}^{n}\). We use the Euclidean norm, i.e., \(\Vert x\Vert :=\sqrt{\langle x,x\rangle }\) for each \(x \in {\mathbb {R}}^{n}\). Let \({\mathbb {N}}\) be the set of natural numbers. Given a nonempty set \(S\subseteq {\mathbb {R}}^{n}\), we denote by clS and convS the closure and the convex hull of the set S,  respectively. The cone generated by \(S\subseteq {\mathbb {R}}^{n}\) is denoted by cone(S),  and is defined as follows:

$$\begin{aligned} cone(S):=\{\lambda s: \lambda \ge 0, \ s \in S\}. \end{aligned}$$

If \( S_{1}, S_{2}, \ldots , S_{m}\) are nonempty subsets of \({\mathbb {R}}^{n}\), the sum of sets, \(\displaystyle {\sum \nolimits _{i=1}^{m}} S_{i},\) is defined as follows:

$$\begin{aligned}{} & {} \displaystyle {\sum _{i=1}^{m}} S_{i}:=\left\{ s_1 + s_2 + \cdots + s_m: s_{i} \in S_{i}, \ i=1,2,\ldots ,m \right\} . \end{aligned}$$

The sublevel set of a function \(f:{\mathbb {R}}^{n} \longrightarrow {\mathbb {R}}\) at the height \(\rho \in {\mathbb {R}}\) [4] is defined by:

$$\begin{aligned} lev_{\le \rho }f:=\{x \in {\mathbb {R}}^{n}: f(x)\le \rho \}. \end{aligned}$$

We also define the sublevel set of the function \(f:{\mathbb {R}}^{n}\longrightarrow {\mathbb {R}}\) at a point \({\bar{x}}\in {\mathbb {R}}^{n}\) [4] as follows:

$$\begin{aligned} lev_{{\bar{x}}}f:=\{x\in {\mathbb {R}}^{n}: f(x)\le f({\bar{x}})\}. \end{aligned}$$

Throughout the paper, we assume that the set F defined by (1.1) is nonempty and closed.

For a nonempty subset S of \({\mathbb {R}}^{n},\) the negative polar cone (dual cone) of S [4, 6] is defined by:

$$\begin{aligned} S^{\circ }:=\{ u \in {\mathbb {R}}^{n}: \langle u, z \rangle \le 0, \ \forall \ z \in S\}, \end{aligned}$$
(2.1)

and the convex cone \( S^{\circ \circ }:=(S^{\circ })^{\circ }\) is called the bipolar of S.

For a nonempty subset S of \( {\mathbb {R}}^{n}\) and a point \( x \in S,\) we recall [1, 4] the definition of the contingent cone as follows:

The contingent cone of S at a point \( x\in S\) is denoted by T(S; x),  and is defined as follows:

$$\begin{aligned}{} & {} T(S;x) \nonumber \\{} & {} :=\{y \in {\mathbb {R}}^{n}: \exists \ \lambda _{m} \in {\mathbb {R}}_{++}, y_{m} \in {\mathbb {R}}^{n} \ s.t. \ \lambda _{m}\longrightarrow 0^{+}, y_{m}\longrightarrow y, x+\lambda _{m}y_{m} \in S, \forall \ m \ge 1 \}, \nonumber \\ {}{} & {} \end{aligned}$$
(2.2)

where the set \({\mathbb {R}}_{++}\) is defined by:

$$\begin{aligned} {\mathbb {R}}_{++}:=\{\lambda \in {\mathbb {R}}: \lambda > 0 \}. \end{aligned}$$

It is easy to show that the contingent cone T(S; x) can be presented as follows:

$$\begin{aligned} T(S;x) =\{y \in {\mathbb {R}}^{n}: \exists \ \lambda _{m} \in {\mathbb {R}}_{++}, x_{m} \in S \ s.t. \ \lambda _{m}\longrightarrow 0^{+}, x_{m}\longrightarrow x \ \text{ and } \ \frac{x_{m}-x}{\lambda _{m}}\longrightarrow y \}. \end{aligned}$$

We define [1] the normal cone to the set \(S \subseteq {\mathbb {R}}^n\) at a point \(x \in S\) by:

$$\begin{aligned} N(S; x): =(T(S; x))^\circ . \end{aligned}$$
(2.3)

Moreover, if S is a convex set, then one can show that (also, see [1, 4]) the normal cone to S at the point \(x \in S\) is equal to the negative polar cone of S at x,  i.e.,

$$\begin{aligned} N(S;x)= (S - x)^\circ :=\{ u \in {\mathbb {R}}^{n}: \langle u,z-x\rangle \le 0, \ \forall \ z \in S\}. \end{aligned}$$
(2.4)

We recall [1] that the set S is regular at a point \(x \in S,\) if S satisfies (2.3).

The following theorem is well known and can be found, for example, in [4, 6].

Theorem 2.1

Let S be a nonempty subset of \({\mathbb {R}}^{n}.\) Then, \(S^{\circ \circ }=cl(conv(cone(S))).\)

In the following, we present the properties of the negative polar cone and the contingent cone.

Lemma 2.2

[6, 7] Let S, \( S_{1}\) and \( S_{2}\) be nonempty subsets of \( {\mathbb {R}}^{n}.\) Then,

  1. (i)

    \(S^{\circ }\) is a nonempty closed convex cone.

  2. (ii)

    \(S^{\circ }=(cl(S))^{\circ }.\)

  3. (iii)

    \(S^{\circ }=(cone(S))^{\circ }=(cl(cone(S)))^{\circ }.\)

  4. (iv)

    If S is a closed convex cone, then, \( S^{\circ \circ }=S.\)

  5. (v)

    If \(S_{1} \subseteq S_{2}\), then, \( S_{1}^{\circ }\supseteq S_{2}^{\circ }.\)

Theorem 2.3

[4] Let S be a nonempty subset of \( {\mathbb {R}}^{n},\) and let \(x \in S.\) Then, the following assertions hold.

  1. (i)

    T(S; x) is a closed cone.

  2. (ii)

    \(T(S;x) \subseteq cl(cone(S- x)).\)

We now recall the concept of the conical hull intersection property (CHIP in short), which is often used in optimization and approximation theory. The CHIP was intensively studied and applied in the literature for the case of finite and infinite intersection of convex or nonconvex sets, for example; see [5,6,7, 13,14,15] and the references therein.

Definition 2.4

[13,14,15]

A collection \(\{C_{1},C_{2},\ldots ,C_{m}\}\) of nonempty sets in \( {\mathbb {R}}^{n}\) which has a nonempty intersection is said to have the “conical hull intersection property” (CHIP in short) at a point \( x \in \bigcap _{j=1}^{m}C_{j},\) if

$$\begin{aligned} T\left( \bigcap _{j=1}^{m}C_{j};x\right) =\bigcap _{j=1}^{m}T(C_{j};x). \end{aligned}$$

Also, [5,6,7] a collection \(\{C_{1},C_{2},\ldots ,C_{m}\}\) of nonempty closed sets in \( {\mathbb {R}}^{n}\) which has a nonempty intersection is said to have the “strong conical hull intersection property” (strong CHIP in short) at a point \( x \in \bigcap _{j=1}^{m}C_{j},\) if

$$\begin{aligned} \left( \displaystyle {\bigcap _{j=1 }^{m}}C_{j}-x \right) ^{\circ }=\displaystyle {\sum _{j=1}^{m}}\Big (C_{j}-x \Big )^{\circ }. \end{aligned}$$

In the following, we give a new definition, we call it the “convex conical hull intersection property” (CCHIP in short).

Definition 2.5

A collection \(\{C_{1},C_{2},\ldots ,C_{m}\}\) of nonempty sets in \( {\mathbb {R}}^{n}\) which has a nonempty intersection is said to have the “convex conical hull intersection property” (CCHIP in short) at a point \( x \in \bigcap _{j=1}^{m}C_{j},\) if

$$\begin{aligned} cl\bigg (conv\bigg (T\bigg (\bigcap _{j=1}^{m}C_{j};x\bigg )\bigg )\bigg )=\bigcap _{j=1}^{m}cl\bigg (conv\bigg (T(C_{j};x)\bigg )\bigg ). \end{aligned}$$

Remark 2.6

Let \(\{C_{1},C_{2},\ldots ,C_{m}\}\) be a collection of nonempty convex sets in \( {\mathbb {R}}^{n}\) with a nonempty intersection. Then the collection \(\{C_{1},C_{2},\ldots ,C_{m}\}\) has the property (CCHIP) at a point \(x \in \bigcap _{j=1}^{m}C_{j}\) if and only if it has the property (CHIP) at the point x. In view of Theorem 2.3(i),  it should be noted that T(S; x) is always closed for any nonempty subset S of \({\mathbb {R}}^n\) and any point \(x \in S.\)

It should be noted that there is no specific relation between the property (CCHIP) and the property (CHIP),  in general. See the following examples. Indeed, in Example 2.7 and Example 4.5 both of the properties (CCHIP) and (CHIP) hold, while in Example 4.7 both of the properties (CCHIP) and (CHIP) do not hold. Moreover, in Example 2.8, the property (CCHIP) does not hold, while the property (CHIP) holds. Furthermore, in Example 2.9, we have shown that the property (CCHIP) may hold even when the property (CHIP) does not hold.

Example 2.7

Let

$$\begin{aligned} C_1:= ({\mathbb {R}}_{+} \times {\mathbb {R}}_{+}) \cup \{(0, -1)\}, \end{aligned}$$

and

$$\begin{aligned} C_2:= ({\mathbb {R}}_{+} \times {\mathbb {R}}_{-}) \cup \{(-1, -1)\} \end{aligned}$$

be two nonconvex sets in \({\mathbb {R}}^2.\)

Put \({\bar{x}}:=(0,0) \in C_1 \cap C_2,\) where one can easily see that

$$\begin{aligned} C_1 \cap C_2 = \{(x_1, x_2) \in {\mathbb {R}}^2: x_1 \ge 0, \ x_2 =0 \} \cup \{(0, -1)\}. \end{aligned}$$

It is clear that \(C_1 \cap C_2\) is not convex, and moreover,

$$\begin{aligned} T(C_1; {{\bar{x}}}) = ({\mathbb {R}}_{+} \times {\mathbb {R}}_{+}) \cup \{(x_1, x_2) \in {\mathbb {R}}^2: x_1 = 0, \ x_2 \le 0 \}, \end{aligned}$$
$$\begin{aligned} T(C_2; {\bar{x}}) = ({\mathbb {R}}_{+} \times {\mathbb {R}}_{-}) \cup \{(x_1, x_2) \in {\mathbb {R}}^2: x_1 = x_2, \ x_1, x_2 \le 0 \}, \end{aligned}$$

and

$$\begin{aligned} T(C_1 \cap C_2; {\bar{x}}) = \{(x_1, x_2) \in {\mathbb {R}}^2: x_1 \ge 0, \ x_2 =0 \} \cup \{(x_1, x_2) \in {\mathbb {R}}^2: x_1 = 0, \ x_2 \le 0 \}, \end{aligned}$$

which are closed, but not convex. Therefore, it is not difficult to show that

$$\begin{aligned} \bigcap _{j=1}^{2} cl\bigg (conv\bigg (T(C_j; {{\bar{x}}})\bigg )\bigg ){} & {} = ({\mathbb {R}}_{+} \times {\mathbb {R}}) \cap \big \{({\mathbb {R}}_{+} \times {\mathbb {R}}_{-}) \\{} & {} \cup \{(x_1, x_2) \in {\mathbb {R}}^2: x_1 \ge x_2, \ x_1, x_2 \le 0\}\big \} \\ {}{} & {} = {\mathbb {R}}_{+} \times {\mathbb {R}}_{-} \\ {}{} & {} = cl\bigg (conv\bigg (T\bigg (\bigcap _{j=1}^{2} C_j; {{\bar{x}}}\bigg )\bigg )\bigg ). \end{aligned}$$

Hence, the collection \(\{C_1, C_2\}\) has the property (CCHIP) at the point \({{\bar{x}}}.\) Moreover, one can easily see that the collection \(\{C_1, C_2\}\) has also the property (CHIP) at the point \({{\bar{x}}}.\)

Example 2.8

Consider the following subsets of \({\mathbb {R}}^2:\)

$$\begin{aligned} C_1{} & {} :=\{(x_{1},x_{2})\in {\mathbb {R}}^{2}: x_{1}\le 0, \ x_{2}> 0, \ x_{1}^{2}\le x_{2} \} \\{} & {} \cup \{(x_{1},x_{2})\in {\mathbb {R}}^{2}: x_{1} < 0, \ x_{2} \le 0 \} \cup \{(0, 0)\}, \end{aligned}$$

and

$$\begin{aligned} C_2{} & {} :=\{(x_{1},x_{2})\in {\mathbb {R}}^{2}: x_{1} \ge 0, \ x_{2} \le 0\}\cup \{(x_{1},x_{2})\in {\mathbb {R}}^{2}: x_{1}\le 0, \ x_{2}\ge 0\}, \end{aligned}$$

which are not convex. Therefore,

$$\begin{aligned} C_1 \cap C_2{} & {} = \{(x_{1},x_{2})\in {\mathbb {R}}^{2}: x_{1}\le 0, \ x_{2} > 0, \ x_{1}^{2}\le x_{2} \}\\{} & {} \cup \{(x_{1},x_{2}) \in {\mathbb {R}}^{2}: x_{1}\le 0, \ x_{2} = 0 \}. \end{aligned}$$

Clearly, \(C_1 \cap C_2\) is not convex. Now, let \({\bar{x}}:=(0, 0) \in C_1 \cap C_2.\) It is easy to check that

$$\begin{aligned} T(C_1 \cap C_2;{\bar{x}}) = {\mathbb {R}}_{-} \times {\mathbb {R}}_{+}. \end{aligned}$$

Moreover, we have

$$\begin{aligned} T(C_1; {\bar{x}}) = {\mathbb {R}}_{-} \times {\mathbb {R}}, \end{aligned}$$

and

$$\begin{aligned} T(C_{2}; {\bar{x}}) = ({\mathbb {R}}_{-} \times {\mathbb {R}}_{+}) \cup ({\mathbb {R}}_{+} \times {\mathbb {R}}_{-}). \end{aligned}$$

Thus, one has

$$\begin{aligned} cl(conv(T(C_1 \cap C_2; {\bar{x}}))){} & {} = {\mathbb {R}}_{-} \times {\mathbb {R}}_{+} \\ {}{} & {} \ne {\mathbb {R}}_{-} \times {\mathbb {R}}\\ {}{} & {} = ({\mathbb {R}}_{-} \times {\mathbb {R}}) \cap {\mathbb {R}}^2 \\ {}{} & {} =cl(conv (T(C_{1}; {\bar{x}}))) \cap cl(conv (T(C_{2}; {\bar{x}}))). \end{aligned}$$

This implies that the collection \(\{C_{1}, C_{2}\}\) does not satisfy the property (CCHIP) at the point \({\bar{x}},\) while the property (CHIP) holds at the point \({\bar{x}}.\) Indeed, one has

$$\begin{aligned} T(C_1; {\bar{x}}) \cap T(C_2; {\bar{x}}) = {\mathbb {R}}_{-} \times {\mathbb {R}}_{+} = T(C_1 \cap C_2;{\bar{x}}). \end{aligned}$$

Example 2.9

Let

$$\begin{aligned} C_1:=\{(-1, 1), (-3, 0), (0, -2), (0, 0)\}, \end{aligned}$$

and

$$\begin{aligned} C_2:=\{(-1, 1), (-2, 0), (0, -2), (0, 0)\} \end{aligned}$$

be two nonconvex sets in \({\mathbb {R}}^2.\) Then,

$$\begin{aligned} C_1 \cap C_2 =\{(-1, 1), (0, -2), (0, 0)\}. \end{aligned}$$

Let \({\bar{x}}:=(0, 0) \in C_1 \cap C_2.\) Therefore, one has

$$\begin{aligned} T(C_1; {\bar{x}}){} & {} = \{(x, y) \in {\mathbb {R}}^2: y =- x, \ x \le 0, \ y \ge 0\} \\ {}{} & {} \cup \{(x, y) \in {\mathbb {R}}^2: x \le 0, \ y = 0\} \\ {}{} & {} \cup \{(x, y) \in {\mathbb {R}}^2: x = 0, \ y \le 0\}, \\ T(C_2; {\bar{x}}){} & {} = \{(x, y) \in {\mathbb {R}}^2: y =- x, \ x \le 0, \ y \ge 0\} \\ {}{} & {} \cup \{(x, y) \in {\mathbb {R}}^2: x \le 0, \ y = 0\} \\ {}{} & {} \cup \{(x, y) \in {\mathbb {R}}^2: x = 0, \ y \le 0\}, \end{aligned}$$
$$\begin{aligned} T(C_1; {\bar{x}}) \cap T(C_2; {\bar{x}}){} & {} = \{(x, y) \in {\mathbb {R}}^2: y =- x, \ x \le 0, \ y \ge 0\} \nonumber \\ {}{} & {} \cup \{(x, y) \in {\mathbb {R}}^2: x \le 0, \ y = 0\} \nonumber \\ {}{} & {} \cup \{(x, y) \in {\mathbb {R}}^2: x = 0, \ y \le 0\}, \end{aligned}$$
(2.5)

and

$$\begin{aligned} T(C_1 \cap C_2; {\bar{x}}){} & {} = \{(x, y) \in {\mathbb {R}}^2: y =- x, \ x \le 0, \ y \ge 0\} \nonumber \\ {}{} & {} \cup \{(x, y) \in {\mathbb {R}}^2: x = 0, \ y \le 0\}. \end{aligned}$$
(2.6)

So, it follows from (2.5) and (2.6) that

$$\begin{aligned} T(C_1; {\bar{x}}) \cap T(C_2; {\bar{x}}) \ne T(C_1 \cap C_2; {\bar{x}}), \end{aligned}$$

and hence, the collection \(\{C_1, C_2\}\) has not the property (CHIP) at the point \({{\bar{x}}}.\) But, we have

$$\begin{aligned} cl\bigg (conv\bigg (T(C_1; {\bar{x}})\bigg )\bigg ) = \{(x, y) \in {\mathbb {R}}^2: y \le -x, \ x\le 0\}, \end{aligned}$$

and

$$\begin{aligned} cl\bigg (conv\bigg (T(C_2; {\bar{x}})\bigg )\bigg ) = \{(x, y) \in {\mathbb {R}}^2: y \le -x, \ x\le 0\}. \end{aligned}$$

Thus, in view of (2.6), we conclude that

$$\begin{aligned} \bigcap _{j=1}^{2} cl\bigg (conv\bigg (T(C_j; {{\bar{x}}})\bigg )\bigg ) = \{(x, y) \in {\mathbb {R}}^2: y \le -x, \ x\le 0\} = cl\bigg (conv\bigg (T\bigg (\bigcap _{j=1}^{2} C_j; {{\bar{x}}}\bigg )\bigg )\bigg ). \end{aligned}$$

This implies that the collection \(\{C_1, C_2\}\) satisfies the property (CCHIP) at the point \({{\bar{x}}}.\)

The definition of the distance function and the projection mapping are given in the following [2, 6].

Definition 2.10

Let C be a nonempty subset of \({\mathbb {R}}^{n}.\) The distance function of C is defined by:

$$\begin{aligned} d_{C}:{\mathbb {R}}^{n} \longrightarrow [0,+\infty ): x \mapsto \inf _{c \in C}\Vert x-c\Vert , \ for \ each \ x \in {\mathbb {R}}^n, \end{aligned}$$

and the projection onto C is the set valued mapping \(P_{C}:{\mathbb {R}}^{n}\rightrightarrows C,\) which is defined at each \(x\in {\mathbb {R}}^{n}\) by:

$$\begin{aligned} P_{C}(x):=argmin_{c\in C}\Vert x-c \Vert :=\{p \in C: \Vert x-p\Vert =d_{C}(x)\}. \end{aligned}$$
(2.7)

If each \(x\in {\mathbb {R}}^{n}\) has at least one best approximation in C, then, C is called a proximinal set. So, C is proximinal if and only if \( P_{C}(x)\ne \emptyset \) for each \(x \in {\mathbb {R}}^{n}.\) It is straightforward to verify that in the finite dimensional case \(P_{C}(x)\) is nonempty for each \(x \in {\mathbb {R}}^{n}\) whenever C is a nonempty closed set in \({\mathbb {R}}^n.\)

The following characterization of the best approximation is well known.

Theorem 2.11

[6]

Let C be a nonempty closed convex subset of \({\mathbb {R}}^{n}\), and let \(x \in {\mathbb {R}}^{n}\) and \(p\in C\). Then, \(p=P_{C}(x)\) if and only if

$$\begin{aligned} \langle x-p, c-p \rangle \le 0, \ \forall \ c \in C. \end{aligned}$$

It is worth noting that the best approximation problem to \(x\in {\mathbb {R}}^{n}\) from the set \(C \subseteq {\mathbb {R}}^n\) can be represented as a minimization problem with the objective function \(\frac{1}{2}\Vert x-\cdot \Vert ^{2}\) over C.

We now give the following necessary condition for optimal solutions of the optimization problem (1.5). Note that if the set F of feasible solutions is nonconvex, then the contingent cone of F can be used as a suitable approximation to the set F.

Theorem 2.12

[4]

Let F be the set of feasible solutions of the optimization problem (1.5) with the objective function \(f:{\mathbb {R}}^n \longrightarrow {\mathbb {R}}\) is smooth, and let \(x_{0} \in F.\) Assume that \(x_{0}\) is an optimal solution of the optimization problem (1.5). Then,

$$\begin{aligned} 0 \in \nabla f(x_{0})+(T(F;x_{0}))^{\circ }. \end{aligned}$$

3 Dual cone characterizations of the normal cone of the feasible set F

In this section, we present a dual cone characterization of the normal cone of the feasible set F. We start with the following definition, which is the definition of pseudoconvexity at a given point [8].

Definition 3.1

Let S be a nonempty subset of \({\mathbb {R}}^{n},\) and let T(S; x) be the contingent cone of S at a point \(x \in S.\)

  1. (i)

    [8] The set S is said to be pseudoconvex at x with respect to T(S; x) if

    $$\begin{aligned} S - x \subseteq T(S;x). \end{aligned}$$
  2. (ii)

    Let K be an arbitrary subset of \({\mathbb {R}}^n\) such that \(S\cap K \ne \emptyset .\) Then, S is said to be pseudoconvex at \(x \in S \cap K\) with respect to \(T(S \cap K;x)\) if

    $$\begin{aligned} S - x \subseteq T(S \cap K;x). \end{aligned}$$

Clearly, if S is pseudoconvex in the sense of (ii),  then by letting \(K:={\mathbb {R}}^n,\) one has S is pseudoconvex in the sense of (i). It is obvious that each convex set is pseudoconvex at each point with respect to the classical contingent cone.

Lemma 3.2

Let S be a nonempty subset of \({\mathbb {R}}^{n},\) and let K be an arbitrary subset of \({\mathbb {R}}^n\) such that \(S\cap K \ne \emptyset .\) If S is pseudoconvex at a point \(x \in S \cap K\) with respect to \(T(S\cap K;x),\) then

$$\begin{aligned} T(S\cap K;x)=cl(cone(S-x)). \end{aligned}$$

Proof

Since S is pseudoconvex at x with respect to \(T(S\cap K;x),\) it follows from Definition 3.1(ii) that \(S-x \subseteq T(S\cap K;x),\) and so, by using Theorem 2.3(i),  we conclude that

$$\begin{aligned} cl(cone(S-x))\subseteq T(S\cap K;x). \end{aligned}$$
(3.1)

On the other hand, one has

$$\begin{aligned} T(S\cap K;x) \subseteq T(S;x) \cap T(K;x) \subseteq T(S;x). \end{aligned}$$

This together with (3.1) and Theorem 2.3(ii) implies that \(T(S\cap K;x)=cl(cone(S-x)),\) which completes the proof. \(\square \)

Lemma 3.3

Let S be a nonempty subset of \({\mathbb {R}}^{n},\) and let K be an arbitrary subset of \({\mathbb {R}}^n\) such that \(S\cap K \ne \emptyset .\) Assume that S is pseudoconvex at a point \(x \in S \cap K\) with respect to \(T(S \cap K;x).\) Then, \(N(S \cap K;x) = (S - x)^\circ .\)

Proof

Suppose that S is pseudoconvex at x with respect to \(T(S \cap K;x).\) Then, in view of Lemma 3.2, we have

$$\begin{aligned} T(S \cap K;x)=cl(cone(S-x)). \end{aligned}$$

This together with (2.3) and Lemma 2.2(iii) implies that

$$\begin{aligned} N(S \cap K; x){} & {} = (T(S \cap K;x))^{\circ } \\ {}{} & {} =(cl(cone(S-x)))^{\circ }\\{} & {} =(cone(S-x))^{\circ }\\{} & {} =(S-x)^{\circ }. \end{aligned}$$

Hence,

$$\begin{aligned} N(S \cap K;x) = (S - x)^\circ =\{u \in {\mathbb {R}}^n: \langle u, y - x \rangle \le 0, \ \forall \ y \in S \}, \end{aligned}$$

and the proof is complete. \(\square \)

We now present a dual cone characterization of the normal cone of the feasible set F,  which has a crucial role for proving the main results.

Proposition 3.4

Let F be given by (1.1), and let K be an arbitrary subset of \({\mathbb {R}}^n\) such that \(F\cap K \ne \emptyset .\) Let \( x \in F \cap K.\) If F is pseudoconvex at x with respect to \(T(F \cap K;x),\) then,

$$\begin{aligned} (N(F \cap K; x))^\circ = cl(conv(T(F \cap K; x))) = (F - x)^{\circ \circ }. \end{aligned}$$

Proof

Suppose that F is pseudoconvex at x with respect to \(T(F \cap K;x).\) Then, by Lemma 3.3, we have \(N(F \cap K; x) = (F - x)^\circ .\) On the other hand, by (2.3), one has \(N(F \cap K; x) = (T(F \cap K; x))^\circ .\) Therefore, by using these equalities and Theorem 2.1 and the fact that \(T(F \cap K; x)\) is a cone, we conclude that

$$\begin{aligned} (F - x)^{\circ \circ }{} & {} = (N(F \cap K; x))^\circ = (T(F \cap K; x))^{\circ \circ } \\ {}{} & {} = cl(conv(cone(T(F \cap K; x)))) \ \\{} & {} = cl(conv(T(F \cap K; x))). \end{aligned}$$

This completes the proof. \(\square \)

Now by letting \(K:= {\mathbb {R}}^n,\) we obtain from Lemma 3.2, Lemma 3.3 and Proposition 3.4 the following corollaries, respectively.

Corollary 3.5

Let S be a nonempty subset of \({\mathbb {R}}^{n}.\) If S is pseudoconvex at a point \(x \in S\) with respect to T(S; x),  then,

$$\begin{aligned} T(S;x)=cl(cone(S-x)). \end{aligned}$$

Corollary 3.6

Let S be a nonempty subset of \({\mathbb {R}}^{n}.\) Assume that S is pseudoconvex at a point \(x \in S\) with respect to T(S; x). Then, \(N(S;x) = (S - x)^\circ .\)

Corollary 3.7

Let F be given by (1.1), and let \( x \in F.\) If F is pseudoconvex at x with respect to T(F; x),  then,

$$\begin{aligned} (N(F; x))^\circ = cl(conv(T(F; x))) = (F - x)^{\circ \circ }. \end{aligned}$$

4 Main results

In this section, let the feasible set F be as in (1.1), given by:

$$\begin{aligned} F:=\bigcap _{j=1}^{r} F_j, \end{aligned}$$

where \(F_1, F_2, \ldots , F_r\) are called the geometric constraints. We assume that the set F is nonempty, closed and pseudoconvex at a reference feasible point \(x \in F\) (note that F is not necessarily convex).

We now give the definition of generalized convex functions called the functions with pseudoconvex sublevel sets [8].

Let \(f: D \subseteq {\mathbb {R}}^n \longrightarrow {\mathbb {R}}\) be a function. We say that f admits pseudoconvex sublevel sets with respect to the contingent cone \(T(lev_{x}f;x),\) if its sublevel sets are pseudoconvex with respect to \(T(lev_{x}f;x)\) at each point \(x \in D,\) i.e.,

$$\begin{aligned} lev_{x}f - x \subseteq T(lev_{x}f;x), \ \forall \ x \in D. \end{aligned}$$

In the sequel, let K be an arbitrary subset of \({\mathbb {R}}^n\) such that \(F \cap K \ne \emptyset .\) For each \({\bar{x}}\in F \cap K,\) let

$$\begin{aligned} \Xi _{K}({\bar{x}}):=\displaystyle {\sum _{j=1}^{r}} N(F_j \cap K; {\bar{x}}):=\left\{ \displaystyle {\sum _{j=1}^{r}} u_{j}: u_{j} \in N(F_j \cap K;{\bar{x}}), \ j=1,2,\ldots ,r \right\} . \end{aligned}$$
(4.1)

It should be noted that in view of (2.3), the set \(\Xi _{K}({\bar{x}})\) is a convex cone, but is not necessarily closed. In the following, we first give a dual cone characterization of the contingent cone \(T(F \cap K;{\bar{x}})\) at some point \({\bar{x}}\in F \cap K,\) which has a crucial role for proving our main results. It is worth noting that by letting \(K:={\mathbb {R}}^n\) in (4.1), we obtain the following convex cone, we denote it by \(\Xi ({\bar{x}})\) \(({\bar{x}}\in F).\)

$$\begin{aligned} \Xi ({\bar{x}}):=\displaystyle {\sum _{j=1}^{r}} N(F_j; {\bar{x}}):=\left\{ \displaystyle {\sum _{j=1}^{r}} u_{j}: u_{j} \in N(F_j;{\bar{x}}), \ j=1,2,\ldots ,r \right\} . \end{aligned}$$
(4.2)

Theorem 4.1

Let the feasible set F be given by (1.1), and let K be given as in (4.1) such that \(F \cap K \ne \emptyset .\) Let \({\bar{x}}\in F \cap K\) be fixed and arbitrary. Suppose that the geometric constraints \(F_1, F_2, \ldots , F_r\) are nonempty and pseudoconvex at \({\bar{x}}\) with respect to \(T(F_j \cap K;{\bar{x}})\) for each \(j=1,2,\ldots ,r,\) respectively, and the collection \(\{F_j \cap K: j=1,2,\ldots ,r\}\) has the property (CCHIP) at the point \({\bar{x}}.\) Then, \((T(F \cap K;{\bar{x}}))^{\circ }=cl(\Xi _{K}({\bar{x}})),\) where the set \(\Xi _{K}({\bar{x}})\) is defined by (4.1).

Proof

Assume if possible that there exists \(u^{*} \in (T(F \cap K;{\bar{x}}))^{\circ }\) such that \(u^{*}\notin cl(\Xi _{K}({\bar{x}}))\). Since \(cl(\Xi _{K}({\bar{x}}))\) is a nonempty closed convex cone, then by using the separation theorem there exists a nonzero continuous linear functional \( l :{\mathbb {R}}^{n} \longrightarrow {\mathbb {R}}\) such that

$$\begin{aligned} \langle l , u^{*}\rangle > 0 \ \ \text{ and } \ \ \langle l , u\rangle \le 0, \ \forall \ u \in cl(\Xi _{K}({\bar{x}})). \end{aligned}$$
(4.3)

Hence,

$$\begin{aligned} \langle l , u \rangle \le 0, \ \forall \ u \in \Xi _{K}({\bar{x}}), \end{aligned}$$

and so, in view of (4.1), we have

$$\begin{aligned} \langle l , u\rangle \le 0, \ \forall \ u \in N(F_j \cap K;{\bar{x}}), \ \forall \ j=1,2,\ldots ,r. \end{aligned}$$

This implies that

$$\begin{aligned} l \in (N(F_j \cap K;{\bar{x}}))^{\circ }, \ \forall \ j=1,2,\ldots ,r. \end{aligned}$$
(4.4)

On the other hand, by the hypothesis, the set \(F_j\) is pseudoconvex at \({\bar{x}}\) with respect to \(T(F_j \cap K;{\bar{x}})\) for each \(j=1,2,\ldots ,r.\) Therefore, in view of Proposition 3.4, we have

$$\begin{aligned} (N(F_j \cap K;{\bar{x}}))^{\circ }=cl(conv(T(F_j \cap K; {\bar{x}}))), \ \forall \ j=1,2,\ldots ,r. \end{aligned}$$
(4.5)

Thus, it follows from (4.4) and (4.5) that

$$\begin{aligned} l \in \bigcap _{j=1}^{r}cl\bigg (conv\bigg (T(F_j \cap K;{\bar{x}})\bigg )\bigg ). \end{aligned}$$

Since, by the hypothesis, the property (CCHIP) holds at \({\bar{x}},\) in view of Definition 2.5 and (1.1), we conclude that

$$\begin{aligned} l \in \textrm{cl}\,\bigg (conv\bigg (T\bigg (\bigcap _{j=1}^r F_j \cap K; {\bar{x}}\bigg )\bigg )\bigg ) = cl(conv(T(F \cap K;{\bar{x}}))). \end{aligned}$$

This together with the fact that \(u^{*} \in (T(F \cap K;{\bar{x}}))^{\circ }\) implies that \(\langle l , u^{*} \rangle \le 0,\) which contradicts (4.3). So, we obtain that

$$\begin{aligned} (T(F \cap K;{\bar{x}}))^{\circ }\subseteq cl(\Xi _{K}({\bar{x}})). \end{aligned}$$
(4.6)

For the reverse inclusion, in view of (2.3) and the fact that the property (CCHIP) holds at \({\bar{x}},\) we get

$$\begin{aligned} N(F_j \cap K;{\bar{x}}){} & {} =(T(F_j \cap K;{\bar{x}}))^{\circ } \\ {}{} & {} = (cl(conv(T(F_j \cap K;{\bar{x}}))))^{\circ } \\ {}{} & {} \subseteq \bigg (\bigcap _{j=1}^{r} cl\bigg (conv\bigg (T(F_j \cap K;{\bar{x}})\bigg )\bigg )\bigg )^{\circ } \\{} & {} = \bigg (cl\bigg (conv\bigg (T\bigg (\bigcap _{j=1}^{r} F_j \cap K; {\bar{x}}\bigg )\bigg )\bigg )\bigg )^{\circ } \\ {}{} & {} = (cl(conv(T(F \cap K;{\bar{x}}))))^{\circ } \\ {}{} & {} = (T(F \cap K; {\bar{x}}))^\circ , \ \forall \ j=1,2,\ldots ,r. \end{aligned}$$

This together with the fact that \((T(F \cap K; {\bar{x}}))^\circ \) is a convex cone implies that

$$\begin{aligned} \displaystyle {\sum _{j=1}^{r}} N(F_j \cap K;{\bar{x}}) \subseteq \displaystyle {\sum _{j=1}^{r}} (T(F \cap K;{\bar{x}}))^{\circ } \subseteq (T(F \cap K;{\bar{x}}))^{\circ }. \end{aligned}$$

Therefore,

$$\begin{aligned} \Xi _{K}({\bar{x}})=\displaystyle {\sum _{j=1}^{r}} N(F_j \cap K; {\bar{x}}) \subseteq (T(F \cap K;{\bar{x}}))^{\circ }. \end{aligned}$$

Therefore, we conclude that \(cl(\Xi _{K}({\bar{x}}))\subseteq (T(F \cap K;{\bar{x}}))^{\circ },\) which completes the proof. \(\square \)

We now give a dual cone characterization of the contingent cone \(T(F;{\bar{x}})\) at some point \({\bar{x}}\in F.\)

Corollary 4.2

Let the feasible set F be given by (1.1), and let \({\bar{x}}\in F\) be fixed and arbitrary. Suppose that the geometric constraints \(F_1, F_2, \ldots , F_r\) are nonempty and pseudoconvex at \({\bar{x}}\) with respect to \(T(F_j;{\bar{x}})\) for each \(j=1,2,\ldots ,r,\) respectively, and the collection \(\{F_j: j=1,2,\ldots ,r\}\) has the property (CCHIP) at the point \({\bar{x}}.\) Then, \((T(F;{\bar{x}}))^{\circ }=cl(\Xi ({\bar{x}})),\) where the set \(\Xi ({\bar{x}})\) is defined by (4.2).

Proof

Under the hypotheses of Corollary 4.2 and by letting \(K:={\mathbb {R}}^n,\) we observe that all hypotheses of Theorem 4.1 hold. Then the result follows from Theorem 4.1. \(\square \)

In the following, we now present necessary and sufficient conditions for characterizing “perturbation property” of the constrained best approximation with geometric constraints.

Theorem 4.3

(Perturbation Property). Let the feasible set F be given by (1.1), and let K be a nonempty closed convex set in \({\mathbb {R}}^{n}\) such that \(F\cap K \ne \emptyset .\) Let \({\bar{x}}\in F \cap K.\) Assume that \(T(F; {\bar{x}})\) and \(T(F \cap K; {\bar{x}})\) are convex sets. Let \(x \in {\mathbb {R}}^{n}\) be fixed and arbitrary. Suppose that the geometric constraints \(F_1, F_2, \ldots , F_r\) are nonempty and pseudoconvex at \({\bar{x}}\) with respect to \(T(F_j \cap K;{\bar{x}})\) for each \(j=1,2,\ldots ,r,\) respectively, and the collection \(\{F_j \cap K: j=1,2,\ldots ,r\}\) has the property (CCHIP) at the point \({\bar{x}}.\) Consider the following assertions:

  1. (i)

    \({\bar{x}}=P_{K}(x-x^{*})\) for some \(x^{*} \in cl(\Xi _{K}({\bar{x}})).\)

  2. (ii)

    \({\bar{x}}=P_{F\cap K}(x).\)

Then, \((i) \Longrightarrow (ii).\) Furthermore, if we assume that the pair \(\{F, K\}\) has the strong CHIP property at the point \({\bar{x}},\) then, \((ii) \Longrightarrow (i).\)

Proof

First note that since F and K are nonempty closed sets, then, \(F\cap K\) is a closed set in \({\mathbb {R}}^n,\) and hence, \(P_{F\cap K}(x)\) and \(P_K(x)\) are nonempty sets for each \(x \in {\mathbb {R}}^n.\)

\([(i) \Longrightarrow (ii)].\) Suppose that (i) holds, i.e., there exists \(x^{*} \in cl(\Xi _{K}({\bar{x}}))\) such that \({\bar{x}}=P_{K}(x-x^{*}).\) We now have \({\bar{x}}\in K\) and K is a nonempty closed convex set in \({\mathbb {R}}^n,\) then in view of Theorem 2.11, we obtain that

$$\begin{aligned}{} & {} \langle x-x^{*}-{\bar{x}}, y-{\bar{x}}\rangle \le 0, \ \forall \ y \in K,\nonumber \\{} & {} \, \Longrightarrow \,x-{\bar{x}}- x^{*} \in (K-{\bar{x}})^{\circ },\nonumber \\{} & {} \, \Longrightarrow \,\exists \ u^{*} \in (K-{\bar{x}})^{\circ } \ \text{ such } \text{ that } \ {\bar{x}}-x=-(u^{*}+x^{*}). \end{aligned}$$
(4.7)

Since \(u^{*} \in (K-{\bar{x}})^{\circ }\), we have

$$\begin{aligned} \langle u^{*}, y-{\bar{x}}\rangle \le 0, \ \forall \ y \in K. \end{aligned}$$
(4.8)

On the other hand, we have \( x^{*} \in cl(\Xi _{K}({\bar{x}})),\) then there exists a sequence \(\{x^{*}_{m}\}_{m \in {\mathbb {N}}} \subset \Xi _{K}({\bar{x}})\) such that \(x^{*}_{m} \longrightarrow x^{*}\) as \( m \longrightarrow +\infty .\)

Since \( x^{*}_{m} \in \displaystyle {\sum \nolimits _{j=1}^{r}} N(F_j \cap K;{\bar{x}})\) \((m \in {\mathbb {N}}),\) it follows that there exists \(u_{mj} \in N(F_j \cap K;{\bar{x}})\) \((j=1,2,\ldots ,r)\) such that \(x^{*}_{m}=\displaystyle {\sum \nolimits _{j=1}^{r}} u_{mj}.\) Since \( u_{mj} \in N(F_j \cap K;{\bar{x}})\) for all \( m \in {\mathbb {N}}\) and all \(j=1,2,\ldots ,r,\) then, \(\langle u_{mj}, y - {\bar{x}}\rangle \le 0\) for all \(y \in F_j \cap K\) and all \(j=1,2,\ldots ,r.\) Note that we used the fact that \(N(F_j \cap K; {\bar{x}}) = (F_j \cap K - {\bar{x}})^\circ .\) Indeed, this equality holds because by the hypothesis, the geometric constraints \(F_1, F_2, \ldots , F_r\) are nonempty and pseudoconvex at \({\bar{x}}\) with respect to \(T(F_j \cap K;{\bar{x}})\) for each \(j=1,2,\ldots ,r,\) and so, by Definition 3.1(ii),  \(F_j - {\bar{x}}\subseteq T(F_j \cap K; {\bar{x}})\) for each \(j=1,2,\ldots , r.\) This implies that \(F_j \cap K - {\bar{x}}\subseteq F_j - {\bar{x}}\subseteq T(F_j \cap K; {\bar{x}})\) for each \(j=1,2,\ldots , r,\) i.e., the sets \(F_j \cap K\) are pseudoconvex at \({\bar{x}}\) with respect to \(T(F_j \cap K;{\bar{x}})\) for each \(j=1,2,\ldots ,r\) (see Definition 3.1(i)). This together with Corollary 3.6 implies that \(N(F_j \cap K; {\bar{x}}) = (F_j \cap K - {\bar{x}})^\circ .\) Thus, we get

$$\begin{aligned} \langle x^{*}_{m}, y-{\bar{x}}\rangle{} & {} =\left\langle \displaystyle {\sum _{j=1}^{r}} u_{mj}, y -{\bar{x}}\right\rangle \\ {}{} & {} = \displaystyle {\sum _{j=1}^{r}} \langle u_{mj}, y-{\bar{x}}\rangle \le 0, \ \forall \ y \in F_{j} \cap K,\ \forall \ j=1,2,\ldots ,r, \ \forall \ m \in {\mathbb {N}}. \end{aligned}$$

This implies that

$$\begin{aligned} \langle x^{*}_{m}, y-{\bar{x}}\rangle \le 0, \ \forall \ y \in F \cap K, \ \forall \ m \in {\mathbb {N}}. \end{aligned}$$

Therefore, since \(x^{*}_{m} \longrightarrow x^{*}\) as \(m \longrightarrow +\infty ,\) it follows that

$$\begin{aligned} \langle x^{*}, y - {\bar{x}}\rangle \le 0, \ \forall \ y \in F \cap K. \end{aligned}$$
(4.9)

Thus, in view of (4.7), (4.8) and (4.9), we conclude that

$$\begin{aligned} \langle {\bar{x}}-x, y- {\bar{x}}\rangle = \langle -(u^{*}+x^{*}), y-{\bar{x}}\rangle \ge 0, \ \forall \ y \in F\cap K. \end{aligned}$$

This implies that

$$\begin{aligned} \frac{1}{2}\Vert x-{\bar{x}}\Vert ^{2}\le \frac{1}{2}\Vert y-x\Vert ^{2}, \ \forall \ y \in F\cap K, \end{aligned}$$

and so, \({\bar{x}}\) is an optimal solution of the optimization problem \(\min _{y \in F\cap K}\frac{1}{2}\Vert y-x\Vert ^{2}.\) Equivalently, \(\Vert x-{\bar{x}}\Vert =\min _{y \in F\cap K}\Vert y-x\Vert \), i.e., \({\bar{x}}= P_{F \cap K}(x)\). Hence, (ii) holds.

\([(ii) \Longrightarrow (i)].\) Suppose that the pair \(\{F,K\}\) has the strong CHIP property at the point \({\bar{x}}.\) Let (ii) hold, i.e., \({\bar{x}}=P_{F\cap K}(x).\) Thus, in view of Definition 2.10, \({\bar{x}}\) is an optimal solution of the optimization problem

$$\begin{aligned} \min _{y \in F \cap K}\frac{1}{2}\Vert y-x\Vert ^{2}. \end{aligned}$$

Since the objective function \(f(\cdot ):=\frac{1}{2}\Vert \cdot -x\Vert ^{2}\) is a smooth convex function with the derivative at \({\bar{x}},\) \(\nabla f({\bar{x}})={\bar{x}}-x,\) then, by using Theorem 2.12, we have

$$\begin{aligned} 0 \in {\bar{x}}-x +(T(F \cap K;{\bar{x}}))^{\circ }. \end{aligned}$$
(4.10)

But, by the hypothesis, one has that the geometric constraints \(F_1, F_2, \ldots , F_r\) are nonempty and pseudoconvex at \({\bar{x}}\) with respect to \(T(F_j \cap K;{\bar{x}})\) for each \(j=1,2,\ldots ,r,\) respectively, and the collection \(\{F_{j} \cap K: j=1,2,\ldots ,r\}\) has the property (CCHIP) at the point \({\bar{x}},\) and moreover, \(T(F \cap K; {\bar{x}})\) is convex. Thus, by Definition 3.1(ii),  we have

$$\begin{aligned} F_{j} - {\bar{x}}\subseteq T(F_{j} \cap K;{\bar{x}}), \ \forall \ j = 1,2,\ldots , r. \end{aligned}$$
(4.11)

This implies that

$$\begin{aligned} (F_{j} \cap K) - {\bar{x}}\subseteq F_{j} - {{\bar{x}}} \subseteq T(F_{j} \cap K;{\bar{x}}), \ \forall \ j = 1,2,\ldots , r. \end{aligned}$$

It follows that

$$\begin{aligned} \left( \bigcap _{j=1}^{r} F_{j} \cap K) - {\bar{x}}\subseteq \bigcap _{j=1}^{r} T(F_{j} \cap K; {\bar{x}}\right) , \end{aligned}$$

and so,

$$\begin{aligned} F \cap K-{\bar{x}}{} & {} \subseteq \bigcap _{j=1}^{r} cl\bigg (conv\bigg (T(F_{j} \cap K; {\bar{x}})\bigg )\bigg ) \nonumber \\{} & {} = cl\bigg (conv\bigg (T\bigg (\bigcap _{j=1}^{r} F_{j} \cap K; {\bar{x}}\bigg )\bigg )\bigg ) \nonumber \\{} & {} = cl(conv(T(F \cap K; {\bar{x}}))) \nonumber \\ {}{} & {} =T(F \cap K; {\bar{x}}). \end{aligned}$$
(4.12)

(Note that \(T(F \cap K; {\bar{x}})\) is closed.) Therefore, (4.12) together with Definition 3.1(i) implies that \(F \cap K\) is pseudoconvex at \({\bar{x}}\) with respect to \(T(F \cap K; {\bar{x}}).\) Hence, in view of (2.3) and Corollary 3.6, one has

$$\begin{aligned} (T(F \cap K; {\bar{x}}))^\circ = N(F \cap K; {\bar{x}}) = (F \cap K - {\bar{x}})^\circ . \end{aligned}$$
(4.13)

Furthermore, in view of the hypothesis, the convexity of \(T(F; {\bar{x}})\) together with (4.11) and the fact that the property (CCHIP) holds at \({\bar{x}}\) implies that

$$\begin{aligned} \bigcap _{j=1}^{r} F_{j} - {\bar{x}}\subseteq \bigcap _{j=1}^{r} T(F_{j} \cap K; {\bar{x}}), \end{aligned}$$

and so,

$$\begin{aligned} F -{\bar{x}}{} & {} \subseteq \bigcap _{j=1}^{r} cl\bigg (conv\bigg (T(F_{j} \cap K; {\bar{x}})\bigg )\bigg ) \\{} & {} = cl\bigg (conv\bigg (T\bigg (\bigcap _{j=1}^{r} F_{j} \cap K; {\bar{x}}\bigg )\bigg )\bigg ) \\{} & {} \subseteq cl\bigg (conv\bigg (T\bigg (\bigcap _{j=1}^{r} F_{j}; {\bar{x}}\bigg ) \cap T(K; {\bar{x}})\bigg )\bigg ) \\ {}{} & {} \subseteq cl\bigg (conv\bigg (T\bigg (\bigcap _{j=1}^{r} F_{j}; {\bar{x}}\bigg )\bigg )\bigg )\\{} & {} = cl(conv(T(F; {\bar{x}}))) \\ {}{} & {} =T(F; {\bar{x}}), \ \text{ note } \text{ that } \ T(F; {\bar{x}}) \ \text{ is } \text{ closed }. \end{aligned}$$

This together with Definition 3.1(i) implies that F is pseudoconvex at \({\bar{x}}\) with respect to \(T(F; {\bar{x}}).\) Hence, in view of Corollary 3.6, one has

$$\begin{aligned} N(F; {\bar{x}}) = (F - {\bar{x}})^\circ . \end{aligned}$$
(4.14)

On the other hand, by the hypothesis, the pair \(\{F,K\}\) has the strong CHIP property at the point \({\bar{x}},\) i.e.,

$$\begin{aligned} (F \cap K -{\bar{x}})^{\circ }=(K-{\bar{x}})^{\circ }+(F-{\bar{x}})^{\circ }. \end{aligned}$$

This together with (4.14) implies that

$$\begin{aligned} (F \cap K -{\bar{x}})^{\circ }{} & {} =(K-{\bar{x}})^{\circ }+ N(F;{\bar{x}}) \nonumber \\{} & {} \subseteq (K - {\bar{x}})^\circ + N(F \cap K; {\bar{x}}). \end{aligned}$$
(4.15)

Thus, (4.10), (4.13) and (4.15) imply that

$$\begin{aligned} 0 \in {\bar{x}}-x +(K-{\bar{x}})^{\circ }+N(F \cap K;{\bar{x}}). \end{aligned}$$
(4.16)

Since all hypotheses of Theorem 4.1 hold, then in view of (2.3) and Theorem 4.1, one has

$$\begin{aligned} N(F \cap K; {\bar{x}}) = (T(F \cap K; {\bar{x}}))^\circ = cl(\Xi _{K}({\bar{x}})). \end{aligned}$$
(4.17)

Therefore, it follows from (4.16) and (4.17) that

$$\begin{aligned} 0 \in {\bar{x}}- x +(K-{\bar{x}})^{\circ } + cl(\Xi _{K}({\bar{x}})). \end{aligned}$$
(4.18)

Therefore, (4.18) implies that there exists \( x^{*} \in cl(\Xi _{K}({\bar{x}}))\) such that \( x-x^{*}-{\bar{x}}\in (K-{\bar{x}})^{\circ }.\) By Theorem 2.11 and the fact that K is closed convex, we conclude that \( {\bar{x}}=P_{K}(x-x^{*})\) for some \(x^{*} \in cl(\Xi _{K}({\bar{x}})),\) i.e., (i) holds, which completes the proof. \(\square \)

We now give necessary and sufficient conditions for the constrained best approximation to \( x \in {\mathbb {R}}^{n}\) from the feasible set F defined by (1.1).

Theorem 4.4

Let the feasible set F be given by (1.1) and \({\bar{x}}\in F,\) and let \(x \in {\mathbb {R}}^{n}\) be fixed and arbitrary. Suppose that \(T(F; {\bar{x}})\) is convex, and the geometric constraints \(F_1, F_2, \ldots , F_r\) are nonempty and pseudoconvex at \({\bar{x}}\) with respect to \(T(F_j;{\bar{x}})\) for each \(j=1,2,\ldots ,r,\) respectively, and the collection \(\{F_{j}: j=1,2,\ldots ,r\}\) has the property (CCHIP) at the point \({\bar{x}}.\) Then, \({\bar{x}}= P_{F}(x)\) if and only if

$$\begin{aligned} 0\in {\bar{x}}- x +cl(\Xi ({\bar{x}})), \end{aligned}$$
(4.19)

where the set \(\Xi ({\bar{x}})\) is defined by (4.2).

Proof

Under the hypotheses of Theorem 4.4 and by letting \(K:={\mathbb {R}}^n,\) one observes that all hypotheses of Theorem 4.3 hold. In this case, we have \((K-{\bar{x}})^{\circ }=\{0\}.\) Therefore, obviously, \(\{F, K\}\) has the strong CHIP property at the point \({\bar{x}}.\)

Now, suppose that \({\bar{x}}=P_{F}(x) = P_{F \cap K}(x),\) i.e., the assertion (ii) in Theorem 4.3 holds. Since \(\{F, K\}\) has the strong CHIP property at the point \({\bar{x}},\) it follows from Theorem 4.3 (the implication \([(ii) \Longrightarrow (i)])\) with \(K = {\mathbb {R}}^n\) that there exists \(x^* \in cl(\Xi _{K}({\bar{x}})) = cl(\Xi ({\bar{x}}))\) such that \({\bar{x}}= P_{K}(x - x^*).\) Hence, in view of Theorem 2.11,

$$\begin{aligned} x - x^* - {\bar{x}}\in (K - {\bar{x}})^\circ = \{0\}, \ \text{ for } \text{ some } \ x^* \in cl(\Xi ({\bar{x}})). \end{aligned}$$

This together with the fact that \(x^* \in cl(\Xi ({\bar{x}}))\) implies that

$$\begin{aligned} 0 \in {\bar{x}}- x + cl(\Xi ({\bar{x}})). \end{aligned}$$

Conversely, assume that \(0 \in {\bar{x}}- x +cl(\Xi ({\bar{x}})).\) This implies that there exists \(x^* \in cl(\Xi ({\bar{x}}))=cl(\Xi _{K}({\bar{x}}))\) such that

$$\begin{aligned} x - x^* - {\bar{x}}=0 \in \{0\} = (K - {\bar{x}})^\circ . \end{aligned}$$

Therefore, in view of Theorem 2.11, \({\bar{x}}= P_K(x - x^*)\) for some \(x^* \in cl(\Xi _{K}({\bar{x}})),\) i.e., the assertion (i) in Theorem 4.3 holds with \(K = {\mathbb {R}}^n.\) Then, by Theorem 4.3 (the implication \([(i) \Longrightarrow (ii)]),\) one has \({\bar{x}}= P_{F \cap K}(x) = P_F(x),\) which completes the proof. \(\square \)

Now, by the following numerical examples we illustrate our results.

Example 4.5

Let

$$\begin{aligned} F_{1}{} & {} :=\left\{ (x_{1},x_{2})\in {\mathbb {R}}^{2}: x_{1}\ge 0, \ x_{2}> 0, \ \frac{1}{2}x_{1}^{3}\le x_{2} \right\} \\ {}{} & {} \cup \{(x_{1},x_{2})\in {\mathbb {R}}^{2}: x_{1} \ge 0, \ x_{2} \le 0 \}, \end{aligned}$$

and

$$\begin{aligned} F_{2}{} & {} :=\{(x_{1},x_{2})\in {\mathbb {R}}^{2}: x_{1}> 0, \ x_{2}\le \sqrt{x_{1}}\}\cup \{(x_{1},x_{2})\in {\mathbb {R}}^{2}: x_{1}\le 0, \ x_{2}\ge 0\}. \end{aligned}$$

It is easy to check that

$$\begin{aligned} F:=F_{1} \cap F_{2}=\left\{ (x_{1},x_{2})\in {\mathbb {R}}^{2}: x_{1}\ge 0, \ x_{2}\ge 0, \ x_{2}\le \sqrt{x_{1}}, \ \frac{1}{2}x_{1}^{3}\le x_{2}\right\} . \end{aligned}$$

It is clear that F is closed. Now, let \({\bar{x}}:=(0,0) \in F.\) Clearly, the geometric constraints \(F_{1}\) and \(F_{2}\) are pseudoconvex at \({\bar{x}}\) with respect to \(T(F_{1};{\bar{x}})\) and \(T(F_{2};{\bar{x}}),\) respectively, but they are not convex. Thus, in view of Corollary 3.5, we have

$$\begin{aligned}{} & {} T(F_{1}; {\bar{x}})=cl(cone(F_{1} - {\bar{x}}))={\mathbb {R}}_{+}\times {\mathbb {R}}, \end{aligned}$$

and

$$\begin{aligned}{} & {} T(F_{2}; {\bar{x}})=cl(cone(F_{2} - {\bar{x}}))={\mathbb {R}}\times {\mathbb {R}}_{+}, \end{aligned}$$

which are closed and convex. Also, by using Corollary 3.5, it is not difficult to show that

$$\begin{aligned} T(F; {\bar{x}})=cl(cone(F - {\bar{x}}))={\mathbb {R}}_{+}\times {\mathbb {R}}_{+}. \end{aligned}$$

Note that \(T(F; {\bar{x}})\) is convex. Hence,

$$\begin{aligned} cl(conv(T(F; {\bar{x}}))){} & {} = T(F; {\bar{x}}) \\{} & {} = {\mathbb {R}}_{+} \times {\mathbb {R}}_{+}\\ {}{} & {} = cl(conv (T(F_{1}; {\bar{x}}))) \cap cl(conv (T(F_{2}; {\bar{x}}))). \end{aligned}$$

This means that the collection \(\{F_{1}, F_{2}\}\) has the property (CCHIP) at the point \({\bar{x}}\in F.\) Therefore, all hypotheses of Corollary 4.2 and Theorem 4.4 hold. Hence, we conclude that the assumptions of Corollary 4.2 and Theorem 4.4 are satisfied as we see in the following (first, note that the property (CHIP) also holds at the point \({\bar{x}}).\) Indeed, in view of (2.3), we have

$$\begin{aligned}{} & {} N(F_{1}; {\bar{x}})=(T(F_{1}; {\bar{x}}))^{\circ }={\mathbb {R}}_{-}\times \{0\},\end{aligned}$$

and

$$\begin{aligned}{} & {} N(F_{2}; {\bar{x}})=(T(F_{2}; {\bar{x}}))^{\circ }=\{0\} \times {\mathbb {R}}_{-}. \end{aligned}$$

Thus,

$$\begin{aligned} \Xi ({\bar{x}})=N(F_{1}; {\bar{x}}) + N(F_{2}; {\bar{x}})= {\mathbb {R}}_{-}\times {\mathbb {R}}_{-}. \end{aligned}$$

Now, let \(x:=(x_{1}, x_2) \in {\mathbb {R}}_{-} \times {\mathbb {R}}_{-}=\Xi ({\bar{x}}).\) If we choose \(x^{*}:=x,\) then, \(x^{*} \in \Xi ({\bar{x}}),\) and so, \( (0,0)={\bar{x}}-x+ x^{*} \in {\bar{x}}- x +\Xi ({\bar{x}})= {\bar{x}}-x +cl(\Xi ({\bar{x}}))\) (note that the set \( \Xi ({\bar{x}})\) is closed). So, by Theorem 4.4, one has, \({\bar{x}}=P_{F}(x)\) for all \(x=(x_1, x_2) \in \Xi ({\bar{x}})=cl(\Xi ({\bar{x}})).\) On the other hand, clearly, \(\Xi (y) = \{(0, 0)\}\) for all \(0 \ne y \in F.\) Thus, \(P_F(y) = y\) for all \(y \in F.\) It should be noted that for all \(y \in {\mathbb {R}}^2 \setminus [F \cup \Xi ({\bar{x}})],\) we have \((0, 0) \notin {\bar{x}}-y +cl(\Xi ({\bar{x}})),\) otherwise there exists \(x^0 \in cl(\Xi ({\bar{x}})) = \Xi ({\bar{x}})\) such that \(x^0 = y \notin \Xi ({\bar{x}}),\) which is a contradiction. Therefore, by Theorem 4.4, \({\bar{x}}\notin P_F(y)\) for all \(y \in {\mathbb {R}}^2 \setminus [F \cup \Xi ({\bar{x}})].\) Also, note that Corollary 4.2 holds because we have

$$\begin{aligned} (T(F; {\bar{x}}))^\circ = {\mathbb {R}}_{-} \times {\mathbb {R}}_{-} = cl(\Xi ({\bar{x}})). \end{aligned}$$

Example 4.6

Let

$$\begin{aligned} F_{1}{} & {} :=\{(x_{1},x_{2})\in {\mathbb {R}}^{2}: x_{1}\le 0, \ x_{2}> 0, \ x_{1}^{2}\le x_{2} \} \\{} & {} \cup \{(x_{1},x_{2})\in {\mathbb {R}}^{2}: x_{1} < 0, \ x_{2} \le 0 \} \cup \{(0, 0)\}, \end{aligned}$$

and

$$\begin{aligned} F_{2}{} & {} := ({\mathbb {R}}_{+} \times {\mathbb {R}}_{-}) \cup ({\mathbb {R}}_{-} \times {\mathbb {R}}_{+}), \end{aligned}$$

which are not convex. One has

$$\begin{aligned} F:=F_{1} \cap F_{2}{} & {} = \{(x_{1},x_{2})\in {\mathbb {R}}^{2}: x_{1}\le 0, \ x_{2} > 0, \ x_{1}^{2}\le x_{2} \}\\{} & {} \cup \{(x_{1},x_{2}) \in {\mathbb {R}}^{2}: x_{1}\le 0, \ x_{2} = 0 \}. \end{aligned}$$

Clearly, F is closed, but not convex. Now, let \({\bar{x}}:=(0, 0) \in F.\) One can easily see that the geometric constraints \(F_{1}\) and \(F_{2}\) are pseudoconvex at \({\bar{x}}\) with respect to \( T(F_{1};{\bar{x}})\) and \(T(F_{2};{\bar{x}})\), respectively. Hence, by Corollary 3.5, it is easy to check that

$$\begin{aligned} T(F_{1}; {\bar{x}}) = cl(cone(F_{1} - {\bar{x}})) = {\mathbb {R}}_{-} \times {\mathbb {R}}, \end{aligned}$$

which is convex, and

$$\begin{aligned} T(F_{2}; {\bar{x}}) = cl(cone(F_{2} - {\bar{x}})) = ({\mathbb {R}}_{-} \times {\mathbb {R}}_{+}) \cup ({\mathbb {R}}_{+} \times {\mathbb {R}}_{-}), \end{aligned}$$

which is not convex. Moreover, in view of Corollary 3.5, we have

$$\begin{aligned} T(F;{\bar{x}}) = cl(cone(F - {\bar{x}})) = {\mathbb {R}}_{-} \times {\mathbb {R}}_{+}. \end{aligned}$$

Note that \(T(F;{\bar{x}})\) is convex. Therefore, one has

$$\begin{aligned} cl(conv(T(F; {\bar{x}}))){} & {} = {\mathbb {R}}_{-} \times {\mathbb {R}}_{+} \\ {}{} & {} \ne {\mathbb {R}}_{-} \times {\mathbb {R}}\\ {}{} & {} = ({\mathbb {R}}_{-} \times {\mathbb {R}}) \cap {\mathbb {R}}^2 \\ {}{} & {} =cl(conv (T(F_{1}; {\bar{x}}))) \cap cl(conv (T(F_{2}; {\bar{x}}))). \end{aligned}$$

This implies that the collection \(\{F_{1}, F_{2}\}\) does not satisfy the property (CCHIP) at the point \({\bar{x}}\in F.\) Note that the property (CHIP) also holds at the point \({\bar{x}}.\) Furthermore, in view of (2.3), one has

$$\begin{aligned}{} & {} N(F_{1}; {\bar{x}})=(T(F_{1}; {\bar{x}}))^{\circ }= {\mathbb {R}}_{+} \times \{0\},\end{aligned}$$

and

$$\begin{aligned}{} & {} N(F_{2}; {\bar{x}})=(T(F_{2}; {\bar{x}}))^{\circ }=\{(0,0)\}, \end{aligned}$$

and hence,

$$\begin{aligned} \Xi ({\bar{x}})=N(F_{1}; {\bar{x}}) + N(F_{2}; {\bar{x}})= {\mathbb {R}}_{+} \times \{0\}. \end{aligned}$$

Clearly, \(\Xi ({\bar{x}})\) is closed. It is obvious that Theorem 4.4 does not hold, because if \(x:=(0, x_2) \in {\mathbb {R}}^2\) with \(x_2 < 0,\) then it is easy to check that \(d_F(x) = \Vert x\Vert = \Vert x - {\bar{x}}\Vert ,\) and so, by Definition 2.10, \({\bar{x}}= P_F(x).\) But, \((0,0)\notin {\bar{x}}- x + cl(\Xi ({\bar{x}})),\) otherwise, one has

$$\begin{aligned} (0, 0) \in {\bar{x}}- x + cl(\Xi ({\bar{x}})) = - x + ({\mathbb {R}}_{+} \times \{0\}). \end{aligned}$$

This implies that \(x \in {\mathbb {R}}_{+} \times \{0\},\) which is a contradiction. Therefore, Theorem 4.4 does not hold. The reason is that the property (CCHIP) does not hold at the point \({\bar{x}}.\) It is worth noting that Corollary 4.2 also does not hold because one has

$$\begin{aligned} (T(F; {\bar{x}}))^\circ = {\mathbb {R}}_{+} \times {\mathbb {R}}_{-} \ne {\mathbb {R}}_{+} \times \{0\} = cl(\Xi ({\bar{x}})). \end{aligned}$$

The reason is that the property (CCHIP) does not hold at the point \({\bar{x}}.\)

Example 4.7

Let

$$\begin{aligned} F_{1}:=\{(x_{1},x_{2})\in {\mathbb {R}}^{2}: x_{1} \ge 0,\ x_{2}^{2}\le x_{1} \}, \end{aligned}$$

and

$$\begin{aligned} F_{2}:=\{(x_{1},x_{2})\in {\mathbb {R}}^{2}: x_{1} \le 0, \ x_{1}\le -x_{2}^{2} \}. \end{aligned}$$

One can see that

$$\begin{aligned} F:=F_{1} \cap F_{2}=\{(0,0)\}. \end{aligned}$$

Therefore, F is closed and the geometric constraints \(F_{1}\) and \(F_{2}\) are convex. Now, let \({\bar{x}}:=(0, 0) \in F.\) So, it is clear that the sets \(F_{1}\) and \(F_{2}\) are pseudoconvex at the point \({\bar{x}}\) with respect to \(T(F_{1};{\bar{x}})\) and \(T(F_{2};{\bar{x}}),\) respectively. Thus, by Corollary 3.5, we have

$$\begin{aligned}{} & {} T(F_{1};{\bar{x}})=cl(cone(F_{1} - {\bar{x}}))={\mathbb {R}}_{+}\times {\mathbb {R}}, \end{aligned}$$

and

$$\begin{aligned}{} & {} T(F_{2};{\bar{x}})=cl(cone(F_{2} -{\bar{x}}))={\mathbb {R}}_{-}\times {\mathbb {R}}, \end{aligned}$$

and also,

$$\begin{aligned} T(F_{1};{\bar{x}}) \cap T(F_{2};{\bar{x}})=\{0\} \times {\mathbb {R}}. \end{aligned}$$

Moreover, in view of Corollary 3.5, the contingent cone \(T(F;{\bar{x}}) = cl(cone(F - {\bar{x}}))=\{(0,0)\},\) which is closed and convex. Hence,

$$\begin{aligned} cl(conv(T(F;{\bar{x}}))){} & {} = T(F; {\bar{x}}) \\ {}{} & {} = \{(0, 0)\} \\ {}{} & {} \ne \{0\} \times {\mathbb {R}}\\ {}{} & {} = cl(conv(T(F_{1};{\bar{x}}))) \cap cl(conv(T(F_{2};{\bar{x}}))), \end{aligned}$$

i.e., the collection \(\{F_{1}, F_{2}\}\) does not satisfy the property (CCHIP) at the point \({\bar{x}}.\) Also, in view of the above, the collection \(\{F_{1}, F_{2}\}\) has not the property (CHIP) at the point \({\bar{x}}.\) Furthermore, in view of (2.3), we have

$$\begin{aligned}{} & {} N(F_{1};{\bar{x}})= (T(F_{1};{\bar{x}}))^{\circ }={\mathbb {R}}_{-} \times \{0\},\end{aligned}$$

and

$$\begin{aligned}{} & {} N(F_{2}; {\bar{x}})= (T(F_{2};{\bar{x}}))^{\circ }={\mathbb {R}}_{+}\times \{0\}, \end{aligned}$$

and so,

$$\begin{aligned} \Xi ({\bar{x}})=N(F_{1};{\bar{x}}) + N(F_{2}; {\bar{x}})={\mathbb {R}}\times \{0\}, \end{aligned}$$

which is closed. Now, we consider \(x:=(0,y) \in {\mathbb {R}}^{2}\) with \(0 \ne y \in {\mathbb {R}}.\) Thus, \((0,0) \notin {\bar{x}}- x +cl(\Xi ({\bar{x}})),\) but clearly, \({\bar{x}}= P_{F}(x).\) Therefore, Theorem 4.4 does not hold. The reason is that the property (CCHIP) does not hold at the point \({\bar{x}}.\) Also, Corollary 4.2 does not hold because

$$\begin{aligned} (T(F; {\bar{x}}))^\circ = {\mathbb {R}}^2 \ne {\mathbb {R}}\times \{0\} = cl(\Xi ({\bar{x}})). \end{aligned}$$

The reason is that the property (CCHIP) does not hold at the point \({\bar{x}}.\)

Remark 4.8

In view of the above examples, one can see that the property (CCHIP) cannot be omitted in Corollary 4.2 and Theorem 4.4. Moreover, it should be noted that Example 4.7 shows that even in the case that the geometric constraints \(F_1, F_2, \ldots , F_r\) are convex, the property (CCHIP) may not be satisfied.

5 Conclusions and applications

In this study, we gave necessary and sufficient conditions for the constrained best approximation problem of an arbitrary point \(x \in {\mathbb {R}}^n\) in the set F,  which is the intersection of the geometric constraints \(F_{1}, F_{2},\ldots , F_{r}\) that are pseudoconvex with respect to the classical contingent cone of \(F_{1}, F_{2},\ldots , F_{r},\) respectively, at a reference feasible point \(x \in F.\) Indeed, we showed under mild conditions on the geometric constraints \(F_{1}, F_{2},\ldots , F_{r}\) that the “perturbation property” of the constrained best approximation from a nonempty closed set \(K \cap F\) is characterized by the “convex conical hull intersection property” (CCHIP in short) at a reference feasible point \(x \in F,\) where K is a nonempty closed convex set in \({\mathbb {R}}^n\) such that \(K \cap F \ne \emptyset .\) We did this by first establishing a dual cone characterization of the contingent cone of the set F. Finally, we presented the “Lagrange multiplier characterizations” of the constrained best approximation. Our results extend and completely solve the constrained best approximation problem, in a more general case, and not only recapture the corresponding known results of [2, 3, 5, 9,10,11,12, 15, 16] and the references therein, but also we can use the new obtained techniques and methods for solving nonconvex and nonsmooth optimization problems with a general objective function other than the norm. These results will appear in a forthcoming study.