A class of nonsmooth fractional multiobjective optimization problems

Abstract

This paper focuses on the study of optimality conditions and duality in nonsmooth fractional multiobjective optimization problems. Applying some advanced tools of variational analysis and generalized differentiation, we establish necessary optimality conditions for (weakly) efficient solutions of a fractional multiobjective optimization problem involving inequality and equality constraints. Sufficient optimality conditions for such solutions to the considered problem are also obtained by means of (strictly) generalized convex-affine functions. In addition, we address a dual problem to the primal one and examine duality relations between them.

1 Introduction

Optimality conditions and duality for (weakly) Pareto/efficient solutions in fractional multiobjective optimization problems have been investigated intensively by many researchers (see e.g., Antczak 2006, 2008; Bahatia and Garg 1998; Bector et al. 1993; Chen 2002; Chinchuluun and Pardalos 2007; Chinchuluun et al. 2007; Kim et al. 2006; Konno and Kuno 1990; Kuk et al. 2001; Lee and Lai 2005; Lai and Ho 2012; Lalitha et al. 2003; Liu 1996; Liu and Yokoyama 1999; Liu and Feng 2007; Long 2011; Nobakhtian 2008; Niculescu 2007; Soleimani-damaneh 2008; Zalmai 2006 and the references therein). One of the main tools used to examine a fractional multiobjective optimization problem is that one employs the separation theorem of convex sets (see e.g., Rockafellar 1970) to provide necessary optimality conditions for (weakly) efficient solutions of the considered problem and exploits various kinds of (generalized) convex/or invex functions to formulate sufficient optimality conditions for such solutions. It should be noted further that since the kinds of (generalized) invex functions mentioned above have been constructed via the convexified/Clarke subdifferential of locally Lipschitz functions, we therefore have to remain using tacitly the separation theorem of convex sets in the proof schemes.

In fact, a characteristic of a fractional multiobjective optimization problem is that its objective function is generally not a convex function. Even under more restrictive concavity/convexity assumptions fractional multiobjective optimization problems are generally nonconvex ones. Besides, the (approximate) extremal principle (Mordukhovich 2006a), which plays a key role in variational analysis and generalized differentiation, has been well-recognized as a variational counterpart of the separation theorem for nonconvex sets. Hence using the extremal principle and other advanced techniques of variational analysis and generalized differentiation to establish optimality conditions seems to be suitable for nonconvex/nonsmooth fractional multiobjective optimization problems.

In this work, we employ some advanced tools of variational analysis and generalized differentiation (e.g., the nonsmooth version of Fermat’s rule, the sum rule and the quotient rule for the limiting/Mordukhovich subdifferential, and the intersection rule for the normal/Mordukhovich cone) to establish necessary optimality conditions for (weakly) Pareto/efficient solutions of a nonsmooth fractional multiobjective optimization problem with inequality and equality constraints. Since the limiting/Mordukhovich subdifferential of a real-valued function at a given point is contained in the convexified/Clarke subdifferential of such a function at the corresponding point (cf. Mordukhovich 2006a), the necessary optimality conditions formulated in terms of the limiting subdifferential are sharper than the corresponding ones expressed in terms of the convexified subdifferential. Sufficient optimality conditions for such solutions to the considered problem are also provided by means of introducing (strictly) generalized convex-affine functions defined in terms of the limiting subdifferential for locally Lipschitz functions. Along with optimality conditions, we state a dual problem to the primal one and explore weak, strong and converse duality relations under assumptions of (strictly) generalized convexity-affineness. Furthermore, examples are given for analyzing and illustrating the obtained results.

In passing, we wish to point out that besides the (weakly) Pareto/efficient solutions, the notion of super minimality/efficiency introduced by Borwein and Zhuang (1993) and more recently investigated by Bao and Mordukhovich (2009) plays also an important role in multiobjective optimization. Since the latter paper successfully established necessary optimality conditions for such efficiency in a general setting by using the above-mentioned generalized differential constructions, it could be possible to obtain results in this vein for fractional multiobjective optimization problems. We leave this for future study.

The rest of the paper is organized as follows. Section 2 contains some basic definitions from variational analysis and several auxiliary results. In Sect. 3, we first establish necessary optimality conditions for (weakly) efficient solutions of a fractional multiobjective optimization problem. Then we supply sufficient optimality conditions for such solutions. Section 4 is devoted to describing duality relations.

2 Preliminaries

Throughout the paper we use the standard notation of variational analysis (see e.g., Mordukhovich 2006a, b). Unless otherwise specified, all spaces under consideration are assumed to be Asplund (i.e., Banach spaces whose separable subspaces have separable duals). The canonical pairing between space X and its topological dual $X^*$ is denoted by $\langle \cdot \,,\cdot \rangle ,$ while the symbol $\Vert \cdot \Vert$ stands for the norm in the considered space. As usual, the polar cone of a set $\Omega \subset X$ is defined by

$$\begin{aligned} \Omega ^\circ :=\{x^*\in X^*\mid \langle x^*,x\rangle \le 0\quad \forall x\in \Omega \}. \end{aligned}$$

(2.1)

Also, for each $m\in \mathbb {N}:=\{1,2,\dots \},$ we denote by $\mathbb {R}^m_+$ the nonnegative orthant of $\mathbb {R}^m.$

Given a multifunction $F:X\rightrightarrows X^*$, we denote by

$$\begin{aligned} \begin{array}{ll} \displaystyle \mathop {\mathrm{Lim}\,\mathrm{sup}}_{x\rightarrow \bar{x}}F(x):=\Big \{x^*\in X^*\big |&{}\exists \; \text{ sequences } \;x_n\rightarrow \bar{x}\; \text{ and } \; x^*_n\xrightarrow {w^*}x^*\\ &{} \text{ with } \;x^*_n\in F(x_n)\; \text{ for } \text{ all } \;n\in \mathbb {N}\Big \} \end{array} \end{aligned}$$

the sequential Painlevé-Kuratowski upper/outer limit of F as $x\rightarrow {\bar{x}},$ where the notation $\xrightarrow {w^*}$ indicates the convergence in the weak$^*$ topology of $X^*.$

A set $\Omega \subset X$ is locally closed if for each $\bar{x}\in \Omega$, there is a neighborhood U of ${\bar{x}}$ such that $\Omega \cap \mathrm{cl\,}U$ is closed. From now on, we always assume that sets under consideration are locally closed. Given $\Omega \subset X$, the regular/Fréchet normal cone to $\Omega$ at $\bar{x}\in \Omega$ is defined by

$$\begin{aligned} {{\widehat{N}}}({\bar{x}};\Omega ):= \Big \{x^*\in X^*\Big |\limsup _ {x\xrightarrow {\Omega }{\bar{x}}}\frac{\langle x^*, x-{\bar{x}}\rangle }{\Vert x-{\bar{x}}\Vert }\le 0\Big \}, \end{aligned}$$

(2.2)

where $x\xrightarrow {\Omega }\bar{x}$ means that $x\rightarrow \bar{x}$ with $x\in \Omega$. If ${\bar{x}}\notin \Omega ,$ we put ${\widehat{N}}({\bar{ x}};\Omega ):=\emptyset .$

The limiting/Mordukhovich normal cone $N({\bar{x}};\Omega )$ at ${\bar{x}}\in \Omega$ is obtained from ${\widehat{N}}(\cdot ;\Omega )$ by taking the sequential Painlevé–Kuratowski upper limits as

$$\begin{aligned} N({\bar{x}};\Omega ):=\mathop {\mathrm{Lim}\,\mathrm{sup}}_{\scriptstyle {x\xrightarrow {\Omega }{\bar{x}}}\scriptstyle {}} {\widehat{N}}(x;\Omega ). \end{aligned}$$

(2.3)

If ${\bar{x}}\notin \Omega ,$ we put $N({\bar{x}};\Omega ):=\emptyset .$

For an extended real-valued function $\varphi :X\rightarrow \overline{\mathbb {R}}:=[-\infty ,\infty ]$, we set

$$\begin{aligned} \hbox {gph}\,\varphi :=\{(x,\mu )\in X\times {\mathbb {R}}\mid \mu =\varphi (x)\},\quad \hbox {epi}\,\varphi :=\{(x,\mu )\in X\times {\mathbb {R}}\; |\; \mu \ge \varphi (x)\}. \end{aligned}$$

The limiting/Mordukhovich subdifferential of $\varphi$ at ${\bar{x}}\in X$ with $|\varphi ({\bar{x}})|<\infty$ is defined by

$$\begin{aligned} \partial \varphi ({\bar{x}})&:=\{ x^*\in X^*\; |\; (x^*,-1)\in N(({\bar{x}},\varphi ({\bar{x}}));\text{ epi }\,\varphi )\}. \end{aligned}$$

(2.4)

If $|\varphi ({\bar{x}})|=\infty ,$ then one puts $\partial \varphi ({\bar{ x}}): =\emptyset .$ It is known (cf. Mordukhovich 2006a) that when $\varphi$ is a convex function, the above-defined subdifferential coincides with the subdifferential in the sense of convex analysis (Rockafellar 1970).

Considering the indicator function $\delta (\cdot ;\Omega )$ defined by $\delta (x;\Omega ):=0$ for $x\in \Omega$ and by $\delta (x;\Omega ):=\infty$ otherwise, we have a relation between the Mordukhovich normal cone and the limiting subdifferential of the indicator function as follows (see Mordukhovich 2006a, Proposition 1.79):

$$\begin{aligned} N({\bar{x}};\Omega )=\partial \delta ({\bar{x}};\Omega )\quad \forall \bar{x}\in \Omega . \end{aligned}$$

(2.5)

The nonsmooth version of Fermat’s rule (see e.g., Mordukhovich 2006a, Proposition 1.114), which is an important fact for many applications, can be formulated as follows: If ${\bar{x}}\in X$ is a local minimizer for $\varphi :X\rightarrow \overline{\mathbb {R}}$, then

$$\begin{aligned} 0\in \partial \varphi ({\bar{x}}). \end{aligned}$$

(2.6)

The following limiting subdifferential sum rule is needed for our study.

Lemma 2.1

(See Mordukhovich 2006a, Theorem 3.36) Let $\varphi _i: X\rightarrow \overline{\mathbb {R}}, i=1,2,\ldots ,n, n\ge 2,$ be lower semicontinuous around ${\bar{x}}\in X,$ and let all these functions except, possibly, one be Lipschitz continuous around ${\bar{ x}}.$ Then one has

$$\begin{aligned} \partial (\varphi _1+\varphi _2+\cdots +\varphi _n)({\bar{x}})\subset \partial \varphi _1({\bar{x}})+\partial \varphi _2({\bar{x}})+\cdots +\partial \varphi _n({\bar{x}}). \end{aligned}$$

(2.7)

Combining this limiting subdifferential sum rule with the quotient rule (cf. Mordukhovich 2006a, Corollary 1.111(ii)), we get an estimate for the limiting subdifferential of quotients.

Lemma 2.2

Let $\varphi _i: X\rightarrow \overline{\mathbb {R}}, i=1,2,$ be Lipschitz continuous around ${\bar{x}}.$ Assume that $\varphi _2({\bar{x}})\ne 0.$ Then one has

$$\begin{aligned} \partial \left( \frac{\varphi _1}{\varphi _2}\right) ({\bar{x}})\subset \frac{ \partial \big (\varphi _2({\bar{x}})\varphi _1\big )({\bar{ x}})+\partial \big (-\varphi _1({\bar{x}})\varphi _2\big )({\bar{ x}})}{[\varphi _2({\bar{x}})]^2}. \end{aligned}$$

(2.8)

Recall Mordukhovich (2006a) that a set $\Omega \subset X$ is sequentially normally compact (SNC) at $\bar{x}\in \Omega$ if for any sequences

$$\begin{aligned} x_k\mathop {\rightarrow }\limits ^{\Omega }\bar{x} \text{ and } x^*_k\mathop {\rightarrow }\limits ^{w^*}0 \text{ with } x^*_k\in {\widehat{N}}_{}(x_k;\Omega ), \end{aligned}$$

one has $\Vert x^*_k\Vert \rightarrow 0$ as $k\rightarrow \infty .$ Obviously, this SNC property is automatically satisfied in finite dimensional spaces. The interested reader is referred to Fabian and Mordukhovich (2003) for a comprehensive comparison of the SNC property and other constructions of this type.

A function $\varphi :X\rightarrow \mathbb {R}$ is called sequentially normally compact (SNC) at ${\bar{x}}\in X$ if $\mathrm{gph\,}\varphi$ is SNC at $({\bar{x}},\varphi ({\bar{x}})).$ According to Mordukhovich (2006a, Corollary 1.69(i)), $\varphi$ is SNC at ${\bar{x}}\in X$ if it is Lipschitz continuous around ${\bar{x}}.$

In what follows, we also need the intersection rule for the normal cones under the fulfillment of the SNC condition.

Lemma 2.3

(See Mordukhovich 2006a, Corollary 3.5) Assume that $\Omega _1, \Omega _2\subset X$ are closed around ${\bar{ x}}\in \Omega _1\cap \Omega _2$ and that at least one of $\{\Omega _1, \Omega _2\}$ is SNC at this point. If

$$\begin{aligned} N({\bar{x}};\Omega _1)\cap \big (- N({\bar{x}};\Omega _2)\big )=\{0\}, \end{aligned}$$

then

$$\begin{aligned} N({\bar{x}};\Omega _1\cap \Omega _2)\subset N({\bar{x}};\Omega _1)+N({\bar{ x}};\Omega _2). \end{aligned}$$

3 Optimality conditions in fractional multiobjective optimization

This section is devoted to studying optimality conditions for fractional multiobjective optimization problems. More precisely, by using the nonsmooth version of Fermat’s rule, the sum rule and the quotient rule for the limiting subdifferentials, and the intersection rule for the Mordukhovich cones, we first establish necessary optimality conditions for (weakly) efficient solutions of a fractional multiobjective optimization problem. Then by imposing assumptions of (strictly) generalized convexity-affineness, we give sufficient optimality conditions for such solutions.

Let $\Omega$ be a nonempty locally closed subset of X,  and let $K:=\{1,\ldots ,m\}, I:=\{1,\ldots ,n\}\cup \emptyset$ and $J:=\{1,\ldots ,l\}\cup \emptyset$ be index sets. In what follows, $\Omega$ is always assumed to be SNC at the point under consideration. This assumption is automatically fulfilled when X is a finite dimensional space.

We consider the following fractional multiobjective optimization problem:

$$\begin{aligned} \mathrm{min}_{\mathbb {R}^m_+}\;\left\{ f(x):=\left( \frac{p_1(x)}{q_1(x)},\cdots , \frac{p_m(x)}{q_m(x)}\right) \,\Big |\,x\in C\right\} , \end{aligned}$$

(P)

where the constraint set C is defined by

$$\begin{aligned} C:=\big \{x\in \Omega \mid \;&g_i(x)\le 0,\, i\in I, h_j(x)=0,\, j\in J\big \}, \end{aligned}$$

(3.1)

and the functions $p_k, q_k, k\in K, g_i, i\in I,$ and $h_j, j\in J$ are locally Lipschitz on X. For the sake of convenience, we further assume that $q_k(x)>0, k\in K$ for all $x\in \Omega$, and that $p_k({\bar{x}})\le 0, k\in K$ for the reference point ${\bar{ x}}\in \Omega$. Also, we use hereafter the notation $g:=(g_1,\ldots ,g_n), h:=(h_1,\ldots ,h_l)$ and $f:=(f_1,\ldots ,f_m)$, where $f_k:=\frac{p_k}{q_k},\, k\in K$.

Definition 3.1

• (i) We say that ${\bar{x}}\in C$ is an efficient solution of problem (P), and write ${\bar{x}}\in \mathcal {S}(P)$, iff

  $$\begin{aligned} \forall x\in C, \quad f(x)-f({\bar{x}})\notin -\mathbb {R}^m_+{\setminus } \{0\}. \end{aligned}$$

• (ii) A point ${\bar{x}}\in C$ is called a weakly efficient solution of problem (P), and write ${\bar{x}}\in \mathcal {S}^{w}(P)$, iff

  $$\begin{aligned} \forall x\in C, \quad f(x)-f({\bar{ x}})\notin -\mathrm{int\,} \mathbb {R}^m_+. \end{aligned}$$

For ${\bar{x}}\in \Omega ,$ let us put

$$\begin{aligned} I({\bar{x}}):=\{i\in I\mid g_i({\bar{x}})=0\},\quad J({\bar{x}}):=\{j\in J\mid h_j({\bar{x}})=0\}. \end{aligned}$$

Definition 3.2

We say that condition (CQ) is satisfied at ${\bar{x}}\in \Omega$ if there do not exist $\beta _i\ge 0, i\in I({\bar{x}})$ and $\gamma _j\ge 0, j\in J({\bar{x}}),$ such that $\sum _{i\in I({\bar{ x}})}\beta _i+\sum _{j\in J({\bar{x}})}\gamma _j\ne 0$ and

$$\begin{aligned} 0\in \sum _{i\in I({\bar{x}})}\beta _i\partial g_i({\bar{x}})+\sum _{j\in J({\bar{x}})}\gamma _j\big (\partial h_j({\bar{x}})\cup \partial (-h_j)({\bar{ x}})\big )+N({\bar{x}};\Omega ). \end{aligned}$$

It is worth to mention here that when considering ${\bar{x}}\in C$ defined in (3.1) with $\Omega =X$ in the smooth setting, the above-defined (CQ) is guaranteed by the Mangasarian-Fromovitz constraint qualification (see e.g., Mordukhovich 2006a for more details).

The following theorem gives a Karush–Kuhn–Tucker (KKT) type necessary optimality condition for (weakly) efficient solutions of problem (P).

Theorem 3.3

Let the (CQ) be satisfied at ${\bar{x}}\in \Omega .$ If ${\bar{x}}\in \mathcal {S}^w(P),$ then there exist $\lambda :=(\lambda _1,\ldots ,\lambda _m)\in \mathbb {R}^m_+\backslash \{0\}, \beta := (\beta _1,\ldots ,\beta _n)\in \mathbb {R}^n_+,$ and $\gamma :=(\gamma _1,\ldots ,\gamma _l)\in \mathbb {R}^l_+$ such that

$$\begin{aligned} 0\in&\sum _{k\in K}\lambda _k\left( \partial p_k({\bar{ x}})-\frac{p_k({\bar{x}})}{q_k({\bar{x}})}\partial q_k({\bar{ x}})\right) +\sum _{i\in I}\beta _i\partial g_i({\bar{x}})\nonumber \\&+\sum _{j\in J}\gamma _j\big (\partial h_j({\bar{x}})\cup \partial (-h_j)({\bar{x}})\big )+N({\bar{x}};\Omega ),\quad \beta _i g_i({\bar{ x}})=0,\quad i\in I. \end{aligned}$$

(3.2)

Proof

Let ${\bar{x}}\in \mathcal {S}^{w}(P)$ and let

$$\begin{aligned} \varphi (x):=\max _{k\in K}{\left\{ \frac{p_k(x)}{q_k(x)}-\frac{p_k({\bar{x}})}{q_k({\bar{ x}})}\right\} }. \end{aligned}$$

We are going to show that

$$\begin{aligned} \varphi ({\bar{x}})\le \varphi (x)\quad \forall x\in C. \end{aligned}$$

(3.3)

Indeed, if this is not the case, then there exists $x_0\in C$ such that $\varphi (x_0)<\varphi ({\bar{x}}).$ Since $\varphi ({\bar{x}})=0,$ it holds that $\max _{k\in K}{\left\{ \frac{p_k(x_0)}{q_k(x_0)}-\frac{p_k({\bar{x}})}{q_k({\bar{ x}})}\right\} }<0.$ Thus,

$$\begin{aligned} f(x_0)-f({\bar{x}})\in -\mathrm{int}\;\mathbb {R}^m_+, \end{aligned}$$

which contradicts the fact that ${\bar{x}}\in \mathcal {S}^{w}(P)$.

We assert by (3.3) that ${\bar{x}}$ is a minimizer of the following scalar problem

$$\begin{aligned} \min _{x\in C}{\varphi (x)}. \end{aligned}$$

Thus, ${\bar{x}}$ is a minimizer of the following unconstrained scalar optimization problem

$$\begin{aligned} \min _{x\in X}{\varphi (x)+\delta (x;C)}. \end{aligned}$$

(3.4)

Applying the nonsmooth version of Fermat’s rule (2.6) to problem (3.4), we have

$$\begin{aligned} 0\in \partial \big (\varphi +\delta (\cdot ;C)\big )({\bar{x}}). \end{aligned}$$

(3.5)

Since the function $\varphi$ is Lipschitz continuous around ${\bar{x}}$ and the function $\delta (\cdot ;C)$ is l.s.c around this point, it follows from the sum rule (2.7) applied to (3.5) and from the relation in (2.5) that

$$\begin{aligned} 0\in \partial \varphi ({\bar{x}})+N({\bar{x}};C). \end{aligned}$$

(3.6)

On the one side, employing the formula for the basic subdifferential of maximum functions (see Mordukhovich 2006a, Theorem 3.46(ii)) and the sum rule (2.7) we obtain

$$\begin{aligned} \partial \varphi ({\bar{ x}})&=\partial \left( \max _{k\in K}{\left\{ \frac{p_k}{q_k}(\cdot )-\frac{p_k({\bar{x}})}{q_k({\bar{ x}})}\right\} }\right) ({\bar{x}})\\&\subset \left\{ \sum _{k\in K}\alpha _k\partial \left( \frac{p_k}{q_k}\right) ({\bar{x}})\,\Big |\, \alpha _k\ge 0, k\in K, \sum _{k\in K}\alpha _k=1\right\} . \end{aligned}$$

Taking (2.8) into account, we arrive at

$$\begin{aligned} \partial \varphi ({\bar{x}})&\subset \left\{ \sum _{k\in K}\alpha _k\frac{ \partial \big (q_k({\bar{ x}})p_k\big )({\bar{x}})+\partial \big (-p_k({\bar{x}})q_k\big )({\bar{ x}})}{[q_k({\bar{x}})]^2}\,\Big |\, \alpha _k\ge 0, k\in K, \sum _{k\in K}\alpha _k=1\right\} \nonumber \\&=\left\{ \sum _{k\in K}\alpha _k\frac{q_k({\bar{x}})\partial p_k({\bar{ x}})-p_k({\bar{x}})\partial q_k({\bar{x}})}{[q_k({\bar{x}})]^2}\,\Big |\, \alpha _k\ge 0, k\in K, \sum _{k\in K}\alpha _k=1\right\} , \end{aligned}$$

(3.7)

where the equality holds due to the fact that $-p_k({\bar{x}})\ge 0, q_k({\bar{x}})>0$ for all $k\in K.$

On the other side, by letting

$$\begin{aligned} \tilde{\Omega }:=\big \{x\in X\mid \;&g_i(x)\le 0,\, i\in I,\\&h_j(x)=0,\, j\in J\big \}, \end{aligned}$$

we have $C=\tilde{\Omega }\cap \Omega .$ The (CQ) being satisfied at ${\bar{x}}$ entails that there do not exist $\beta _i\ge 0, i\in I({\bar{ x}}),$ and $\gamma _j\ge 0, j\in J({\bar{x}})=J$ such that $\sum _{i\in I({\bar{x}})}\beta _j+\sum _{j\in J}\gamma _j\ne 0$ and

$$\begin{aligned} 0\in \sum _{i\in I({\bar{x}})}\beta _i\partial g_i({\bar{x}})+\sum _{j\in J}\gamma _j\big (\partial h_j({\bar{x}})\cup \partial (-h_j)({\bar{ x}})\big ). \end{aligned}$$

Hence, we get by Mordukhovich (2006a, Corollary 4.36) that

$$\begin{aligned} N({\bar{x}};\tilde{\Omega })\subset&\Big \{ \sum _{i\in I({\bar{ x}})}\beta _i\partial g_i({\bar{x}})+\sum _{j\in J}\gamma _j\big (\partial h_j({\bar{x}})\cup \partial (-h_j)({\bar{x}})\big ) \,\Big |\,\beta _i\ge 0, i\in I({\bar{x}}), \nonumber \\&\quad \gamma _j\ge 0, j\in J\Big \}. \end{aligned}$$

(3.8)

As the (CQ) is satisfied at ${\bar{x}}$ and $\Omega$ is assumed to be SNC at this point, we apply Lemma 2.3 to obtain the following

$$\begin{aligned} N({\bar{x}};C)= N({\bar{x}};\tilde{\Omega }\cap \Omega )\subset N({\bar{ x}};\tilde{\Omega })+N({\bar{x}};\Omega ). \end{aligned}$$

(3.9)

It follows from (3.6)–(3.9) that

$$\begin{aligned} 0\in&\left\{ \sum _{k\in K}\frac{\alpha _k}{q_k({\bar{x}})}\left( \partial p_k({\bar{x}})-\frac{p_k({\bar{x}})}{q_k({\bar{x}})}\partial q_k({\bar{ x}})\right) \,\Big |\, \alpha _k\ge 0, k\in K, \sum _{k\in K}\alpha _k=1\right\} \nonumber \\&+ \left\{ \sum _{i\in I({\bar{x}})}\beta _i\partial g_i({\bar{ x}})+\sum _{j\in J}\gamma _j\big (\partial h_j({\bar{x}})\cup \partial (-h_j)({\bar{x}})\big ) \,\Big |\,\beta _i\ge 0, i\in I({\bar{ x}}),\gamma _j\ge 0, j\in J \right\} \nonumber \\&+N({\bar{x}};\Omega ). \end{aligned}$$

(3.10)

Now, put $\beta _i:=0$ for $i\in I{\setminus } I({\bar{x}})$, and let $\lambda _k:=\frac{\alpha _k}{q_k({\bar{x}})}$ for $k\in K.$ It is clear that (3.10) implies (3.2) and so, the proof is complete. $\square$

A simple example below shows that the conclusion of Theorem 3.3 may fail if the (CQ) is not satisfied at the point in question.

Example 3.4

Let $f: \mathbb {R}\rightarrow \mathbb {R}^2$ be defined by

$$\begin{aligned} f(x):=\left( \frac{p_1(x)}{q_1(x)},\frac{p_2(x)}{q_2(x)}\right) , \end{aligned}$$

where $p_1(x)=p_2(x):=x, q_1(x)=q_2(x):=x^2+1, x\in \mathbb {R},$ and let $g,h: \mathbb {R}\rightarrow \mathbb {R}$ be given by $g(x):=x^2, h(x):=0, x\in \mathbb {R}.$ We consider problem (P) with $m:=2$ and $\Omega :=(-\infty ,0]\subset \mathbb {R}$. Then $C=\{0\}$ and thus, ${\bar{x}}:=0\in \mathcal {S}^w(P)(=\mathcal {S}(P)).$ In this setting, we have $N({\bar{ x}};\Omega )=[0,+\infty ).$ Now, we can check that condition (CQ) is not satisfied at ${\bar{x}}.$ Meantime, ${\bar{x}}$ does not satisfy (3.2) either.

We refer the reader to a result (Bao et al. 2007) about necessary optimality conditions for a more general multiobjective fractional program with equilibrium constraints by way of a different approach. More concretely, the paper (Bao et al. 2007) considers the problem (P) with an additional equilibrium constraint:

$$\begin{aligned} 0\in G(x)+Q(x), \end{aligned}$$

where $G, Q:X\rightrightarrows Y$ are multifunctions between Banach spaces. Their approach is to compute the Mordukhovich/limiting coderivative of the epigraphical multifunction of the component functions $\varphi _k:=\frac{p_k}{q_k}, k\in K$ for deriving the optimality conditions. Under a constraint qualification condition (Bao et al. 2007, Theorem 4.2) obtains a KKT type necessary optimality condition for a (local) weakly efficient solution ${\bar{ x}}$ involving the values of the limiting coderivatives of $D^*G({\bar{ x}},{\bar{y}})$ and $D^*Q({\bar{x}},{\bar{y}})$, where ${\bar{y}}\in G({\bar{x}})\cap (-Q({\bar{x}}))$, and of the subdifferentials $\frac{ \partial \big (q_k({\bar{x}})p_k\big )({\bar{x}})+\partial \big (-p_k({\bar{ x}})q_k\big )({\bar{x}})}{[q_k({\bar{x}})]^2}, k\in K.$ Hence, Bao et al. (2007, Theorem 4.2) is more general than Theorem 3.3 due to the former contains the data of equilibrium constraint $D^*G({\bar{x}},{\bar{y}})$ and $D^*Q({\bar{x}},{\bar{ y}})$. However, by exploiting the exclusive structure of our problem, we can elaborate separately the subdifferentials of the functions $p_k$ and $q_k, k\in K$ at the referenced point and turn the necessary optimality criterion into a traditional representation of the KKT condition (cf. 3.2), which allows us to be able to provide sufficient optimality conditions and explore duality relations in the sequel.

The next example illustrates that Theorem 3.3 works better in comparison with some of the existing results about optimality conditions for fractional multiobjective optimization problems, for instance (Kim et al. 2006).

Example 3.5

Let $f: \mathbb {R}\rightarrow \mathbb {R}^2$ be defined by

$$\begin{aligned} f(x):=\left( \frac{p_1(x)}{q_1(x)},\frac{p_2(x)}{q_2(x)}\right) , \end{aligned}$$

where $p_1(x)=p_2(x):=|x|, q_1(x)=q_2(x):=-|x|+1, x\in \mathbb {R},$ and let $g: \mathbb {R}\rightarrow \mathbb {R}$ be given by $g(x):=-x-1, x\in \mathbb {R}.$ Let us consider problem (P) with $K:=\{1,2\}, I:=\{1\}, J:=\emptyset ,$ and $\Omega :=(-1, 1)\subset \mathbb {R}$. It is easy to check that ${\bar{x}}:=0\in \mathcal {S}^w(P)$ and the (CQ) is satisfied at this point. So, in this setting we can apply Theorem 3.3 to conclude that ${\bar{ x}}$ satisfies condition (3.2). Meanwhile, since the functions $q_1, q_2$ are not differentiable at ${\bar{x}}$ (Kim et al. 2006, Theorem 2.2) is not applicable to this problem.

It should be noted further that, in general, a feasible point of problem (P) satisfying condition (3.2) is not necessarily to be a weakly efficient solution even in the smooth case. This will be illustrated by the following example.

Example 3.6

Let $f: \mathbb {R}\rightarrow \mathbb {R}^2$ be defined by

$$\begin{aligned} f(x):=\left( \frac{p_1(x)}{q_1(x)},\frac{p_2(x)}{q_2(x)}\right) , \end{aligned}$$

where $p_1(x)=p_2(x):=x^3-1, q_1(x)=q_2(x):=x^2+1, x\in \mathbb {R},$ and let $g,h: \mathbb {R}\rightarrow \mathbb {R}$ be given by $g(x):=-x^2, h(x):=0, x\in \mathbb {R}.$ Let us consider problem (P) with $m:=2$ and $\Omega :=(-\infty , 1]\subset \mathbb {R}$. Then $C=\Omega$ and thus, ${\bar{x}}:=0\in C$. In this setting, we have $N({\bar{x}};\Omega )=\{0\}.$ Observe that ${\bar{x}}$ satisfies condition (3.2). However, ${\bar{x}}\notin \mathcal {S}^w(P)$.

By virtue of Example 3.6, obtaining sufficient optimality conditions for (weakly) efficient solutions of problem (P) requires concepts of (generalized) convexity-affineness-type for locally Lipschitz functions.

Definition 3.7

• (i) We say that (f, g; h) is generalized convex-affine on $\Omega$ at ${\bar{x}}\in \Omega$ if for any $x\in \Omega , \;u^*_k\in \partial p_k({\bar{x}}),\; v^*_k\in \partial q_k({\bar{x}}), \; k\in K,\; x^*_i\in \partial g_i({\bar{x}}), i\in I,$ and $y^*_j\in \partial h_j({\bar{x}})\cup \partial (-h_j)({\bar{x}}), j\in J$ there exists $\nu \in N({\bar{x}};\Omega )^\circ$ such that

  $$\begin{aligned} p_k(x)-p_k({\bar{x}})\ge&\langle u^*_k,\nu \rangle , k\in K,\\ q_k(x)-q_k({\bar{x}})\ge&\langle v^*_k,\nu \rangle , k\in K,\\ g_i(x)-g_i({\bar{x}})\ge&\langle x^*_i,\nu \rangle , i\in I,\\ h_j(x)-h_j({\bar{x}})=&\omega _j\langle y^*_j,\nu \rangle , j\in J, \end{aligned}$$

  where $\omega _j=1$ (respectively, $\omega _j=-1$) whenever $y^*_j\in \partial h_j({\bar{ x}})$ (respectively, $y^*_j\in \partial (-h_j)({\bar{x}})$).

• (ii) We say that (f, g; h) is strictly generalized convex-affine on $\Omega$ at ${\bar{x}}\in \Omega$ if for any $x\in \Omega \backslash \{{\bar{x}}\}, \; \;u^*_k\in \partial p_k({\bar{x}}), \;v^*_k\in \partial q_k({\bar{x}}),\; k\in K,\; x^*_i\in \partial g_i({\bar{x}}), i\in I,$ and $y^*_j\in \partial h_j({\bar{x}})\cup \partial (-h_j)({\bar{x}}), j\in J$ there exists $\nu \in N({\bar{ x}};\Omega )^\circ$ such that

  $$\begin{aligned} p_k(x)-p_k({\bar{x}})>&\langle u^*_k,\nu \rangle , k\in K,\\ q_k(x)-q_k({\bar{x}})\ge&\langle v^*_k,\nu \rangle , k\in K,\\ g_i(x)-g_i({\bar{x}})\ge&\langle x^*_i,\nu \rangle , i\in I,\\ h_j(x)-h_j({\bar{x}})=&\omega _j\langle y^*_j,\nu \rangle , j\in J, \end{aligned}$$

  where $\omega _j=1$ (respectively, $\omega _j=-1$) whenever $y^*_j\in \partial h_j({\bar{x}})$ (respectively, $y^*_j\in \partial (-h_j)({\bar{x}})$).

It is clear that if $\Omega$ is convex, $p_k, q_k, k\in K, g_i, i\in I$ are convex, and $h_j, j\in J$ are affine, then (f, g; h) is generalized convex-affine on $\Omega$ at ${\bar{x}}\in \Omega$ with $\nu :=x-{\bar{x}}$ for each $x\in \Omega .$ And besides, when $q_k(x)\equiv 1$ for $k\in K$ (i.e., $f:=(p_1,\dots ,p_m)$), the above-defined notions reduce respectively to L-(strictly) invex-infine functions given in Chuong (2012), Chuong and Kim (2014). Hence, the class of generalized convex-affine functions surely contains some nonconvex functions (see e.g., Chuong 2012, Example 3.3). The reader is referred to Chuong (2012, 2013), Chuong and Kim (2014), Sach et al. (2003) for some properties and applications of (L-) invex-infine functions.

We are now in a position to provide sufficient conditions for a feasible point of problem (P) to be a weakly efficient (or efficient) solution.

Theorem 3.8

Assume that ${\bar{x}}\in C$ satisfies condition (3.2).

• (i) If (f, g; h) is generalized convex-affine on $\Omega$ at ${\bar{x}},$ then ${\bar{x}}\in \mathcal {S}^w(P).$

• (ii) If (f, g; h) is strictly generalized convex-affine on $\Omega$ at ${\bar{x}},$ then ${\bar{x}}\in \mathcal {S}(P)$.

Proof

Since ${\bar{x}}$ satisfies condition (3.2), there exist $\lambda :=(\lambda _1,\ldots ,\lambda _m)\in \mathbb {R}^m_+{\setminus } \{0\}, \mu :=(\mu _1,\ldots ,\mu _n)\in \mathbb {R}^n_+, \gamma :=(\gamma _1,\ldots ,\gamma _l)\in \mathbb {R}^l_+, u^*_k\in \partial p_k({\bar{x}}), v^*_k\in \partial q_k({\bar{x}}), k\in K, x^*_i\in \partial g_i({\bar{x}}), i\in I$ with $\mu _ig_i({\bar{x}})=0,$ and $y^*_j\in \partial h_j({\bar{x}})\cup \partial (-h_j)({\bar{x}}), j\in J$ such that

$$\begin{aligned} -\left[ \sum _{k\in K}\lambda _k \left( u^*_k-\frac{p_k({\bar{ x}})}{q_k({\bar{x}})}v_k^*\right) +\sum _{i\in I}\mu _i x^*_i+\sum _{j\in J}\gamma _j y^*_j\right] \in N({\bar{x}};\Omega ). \end{aligned}$$

(3.11)

We first justify (i). Assume on the contrary that ${\bar{x}} \notin \mathcal {S}^w(P).$ This means that there is $\hat{x}\in C$ such that

$$\begin{aligned} f(\hat{x})-f({\bar{x}})\in -\mathrm{int\,}\mathbb {R}^m_+. \end{aligned}$$

(3.12)

By the generalized convex-affine property of (f, g; h) on $\Omega$ at ${\bar{x}},$ for $\hat{x}$ above, there exists $\nu \in N({\bar{ x}};\Omega )^\circ$ such that

$$\begin{aligned}&\sum _{k\in K}\lambda _k \left( \langle u^*_k,\nu \rangle -\frac{p_k({\bar{ x}})}{q_k({\bar{x}})}\langle v_k^*,\nu \rangle \right) +\sum _{i\in I}\mu _i \langle x^*_i,\nu \rangle +\sum _{j\in J}\gamma _j \langle y^*_j,\nu \rangle \\&\le \sum _{k\in K}\lambda _k \left[ p_k(\hat{x})-p_k({\bar{ x}})-\frac{p_k({\bar{x}})}{q_k({\bar{x}})}\big ( q_k(\hat{x})-q_k({\bar{ x}})\big )\right] +\sum _{i\in I}\mu _i \big (g_i(\hat{x})-g_i({\bar{ x}})\big )\\&\quad +\sum _{j\in J}\frac{1}{\omega _j}\gamma _j \big (h_j(\hat{x})-h_j({\bar{x}})\big )\\&=\sum _{k\in K}\lambda _k \left( p_k(\hat{x})-\frac{p_k({\bar{ x}})}{q_k({\bar{x}})} q_k(\hat{x})\right) +\sum _{i\in I}\mu _i \big (g_i(\hat{x})-g_i({\bar{x}})\big )\\&\quad +\sum _{j\in J}\frac{1}{\omega _j}\gamma _j \big (h_j(\hat{x})-h_j({\bar{x}})\big ), \end{aligned}$$

where $\omega _j\in \{-1, 1\}, j\in J.$ Due to the definition of polar cone (2.1), it follows from (3.11) and the relation $\nu \in N({\bar{x}};\Omega )^\circ$ that

$$\begin{aligned} 0\le \sum _{k\in K}\lambda _k \left( \langle u^*_k,\nu \rangle -\frac{p_k({\bar{ x}})}{q_k({\bar{x}})}\langle v_k^*,\nu \rangle \right) +\sum _{i\in I}\mu _i \langle x^*_i,\nu \rangle +\sum _{j\in J}\gamma _j \langle y^*_j,\nu \rangle . \end{aligned}$$

(3.13)

Hence,

$$\begin{aligned} 0&\le \sum _{k\in K}\lambda _k \left( p_k(\hat{x})-\frac{p_k({\bar{ x}})}{q_k({\bar{x}})} q_k(\hat{x})\right) +\sum _{i\in I}\mu _i \big (g_i(\hat{x})-g_i({\bar{x}})\big )\nonumber \\&\quad +\sum _{j\in J}\sigma _j \big (h_j(\hat{x})-h_j({\bar{x}})\big ), \end{aligned}$$

(3.14)

where $\sigma _j:=\frac{\gamma _j}{\omega _j}\in \mathbb {R}, j\in J.$ In addition, it holds that

$$\begin{aligned} \sum _{i\in I}\mu _i \big (g_i(\hat{x})-g_i({\bar{x}})\big )+\sum _{j\in J}\sigma _j \big (h_j(\hat{x})-h_j({\bar{x}})\big )\le 0 \end{aligned}$$

due to the fact that $\mu _i g_i({\bar{x}})=0, \mu _ig_i(\hat{x})\le 0, i\in I$, and $h_j({\bar{x}})=0, h_j(\hat{x})=0, j\in J.$ So, we get by (3.14) that

$$\begin{aligned} 0\le \sum _{k\in K}\lambda _k \left( p_k(\hat{x})-\frac{p_k({\bar{ x}})}{q_k({\bar{x}})} q_k(\hat{x})\right) . \end{aligned}$$

This entails that there is $k_0\in K$ such that

$$\begin{aligned} 0\le p_{ k_0}(\hat{x})-\frac{p_{k_0}({\bar{x}})}{q_{k_0}({\bar{x}})} q_{k_0}({\hat{x}}) \end{aligned}$$

(3.15)

due to $\lambda \in \mathbb {R}^m_+{\setminus } \{0\}.$ The inequality in (3.15) is equivalent to the following one

$$\begin{aligned} f_{k_0}({\bar{x}})\le f_{k_0}({\hat{x}}), \end{aligned}$$

which contradicts (3.12) and, therefore, the proof of (i) is complete.

Now, we prove (ii). Suppose to the contrary that ${\bar{x}} \notin \mathcal {S}(P).$ Then there is ${\hat{x}}\in C$ such that

$$\begin{aligned} f({\hat{x}})-f({\bar{x}})\in -\mathbb {R}^m_+\backslash \{0\}. \end{aligned}$$

(3.16)

By the strictly generalized convex-affine property of (f, g; h) on $\Omega$ at ${\bar{x}},$ for ${\hat{x}}$ above, there exists $\nu \in N({\bar{x}};\Omega )^\circ$ such that

$$\begin{aligned}&\sum _{k\in K}\lambda _k \left( \langle u^*_k,\nu \rangle -\frac{p_k({\bar{ x}})}{q_k({\bar{x}})}\langle v_k^*,\nu \rangle \right) +\sum _{i\in I}\mu _i \langle x^*_i,\nu \rangle +\sum _{j\in J}\gamma _j \langle y^*_j,\nu \rangle \\&< \sum _{k\in K}\lambda _k \left[ p_k({\hat{x}})-p_k({\bar{ x}})-\frac{p_k({\bar{x}})}{q_k({\bar{x}})}\big ( q_k({\hat{x}})-q_k({\bar{ x}})\big )\right] +\sum _{i\in I}\mu _i \big (g_i({\hat{x}})-g_i({\bar{ x}})\big )\\&\quad +\sum _{j\in J}\frac{1}{\omega _j}\gamma _j \big (h_j({\hat{x}})-h_j({\bar{x}})\big ), \end{aligned}$$

where $\omega _j\in \{-1, 1\}, j\in J.$ Similar to the proof of (i), we obtain (3.13) and then arrive at

$$\begin{aligned} 0< \sum _{k\in K}\lambda _k \left( p_k({\hat{x}})-\frac{p_k({\bar{ x}})}{q_k({\bar{x}})} q_k({\hat{x}})\right) . \end{aligned}$$

This entails that there is $k_0\in K$ such that

$$\begin{aligned} 0< p_{ k_0}({\hat{x}})-\frac{p_{k_0}({\bar{x}})}{q_{k_0}({\bar{x}})} q_{k_0}({\hat{x}}). \end{aligned}$$

Equivalently,

$$\begin{aligned} f_{k_0}({\bar{x}})< f_{k_0}({\hat{x}}). \end{aligned}$$

It together with (3.16) gives a contradiction, which completes the proof. $\square$

4 Duality in fractional multiobjective optimization

In this section we propose a dual problem to the primal one in the sense of Mond and Weir (1981) and examine weak, strong, and converse duality relations between them. Note further that another dual problem formulated in the sense of Wolfe (1961) can be similarly treated.

Let $z\in X, \lambda :=(\lambda _1,\dots ,\lambda _m)\in \mathbb {R}^m_+\backslash \{0\}, \mu :=(\mu _1,\dots ,\mu _n)\in \mathbb {R}^n_+,$ and $\gamma :=(\gamma _1,\dots ,\gamma _l)\in \mathbb {R}^l_+$. In connection with the fractional multiobjective optimization problem (P), we consider a fractional multiobjective dual problem of the form:

$$\begin{aligned} \mathrm{max}_{\mathbb {R}^m_+}\;\left\{ {\bar{ f}}(z,\lambda ,\mu ,\gamma ):=\left( \frac{p_1(z)}{q_1(z)},\cdots , \frac{p_m(z)}{q_m(z)}\right) \,\Big |\,(z,\lambda ,\mu ,\gamma )\in C_{D}\right\} . \end{aligned}$$

(D)

Here the constraint set $C_{D}$ is defined by

$$\begin{aligned} C_{D}:=\big \{(z,\lambda ,\mu ,\gamma )&\in \Omega \times (\mathbb {R}^m_+\backslash \{0\})\times \mathbb {R}^n_+\times \mathbb {R}^l_+\mid \; 0\in \sum _{k\in K}\lambda _k\left( \partial p_k(z)-\frac{p_k(z)}{q_k(z)}\partial q_k(z)\right) \\&\quad +\sum _{i\in I}\mu _i\partial g_i(z)+\sum _{j\in J}\gamma _j\big (\partial h_j(z)\cup \partial (-h_j)(z)\big ) +N(z;\Omega ),\\&\quad \langle \mu ,g(z)\rangle +\langle \sigma ,h(z)\rangle \ge 0\quad \forall \sigma \in \mathbb {S}(0,||\gamma ||) \big \}, \end{aligned}$$

where $\mathbb {S}(0,||\gamma ||):=\{\sigma \in \mathbb {R}^l\mid ||\sigma ||=||\gamma ||\}.$

We need to address here that an efficient solution (resp., a weakly efficient solution) of problem (D) is similarly defined as in Definition 3.1 by replacing $-\mathbb {R}^m_+$ (resp., $\mathrm{int\,}\mathbb {R}^m_+$) by $\mathbb {R}^m_+$ (resp., $\mathrm{-int\,}\mathbb {R}^m_+$). Also, we denote the set of efficient solutions (resp., weakly efficient solutions) of problem (D) by $\mathcal {S}(D)$ (resp., $\mathcal {S}^w(D)$).

In what follows, we use the following notation for convenience.

$$\begin{aligned} u\prec v&\Leftrightarrow u-v\in -\mathrm{int\,}\mathbb {R}^m_+,\;u\nprec v \text{ is } \text{ the } \text{ negation } \text{ of } u\prec v,\\ u\preceq v&\Leftrightarrow u-v\in -\mathbb {R}^m_+\backslash \{0\},\;u\npreceq v \text{ is } \text{ the } \text{ negation } \text{ of } u\preceq v. \end{aligned}$$

The first theorem in this section describes weak duality relations between the primal problem (P) and the dual problem (D).

Theorem 4.1

(Weak Duality) Let $x\in C$ and let $(z,\lambda ,\mu ,\gamma )\in C_{D}$.

• (i) If (f, g; h) is generalized convex-affine on $\Omega$ at z,  then

  $$\begin{aligned} f( x)\nprec {\bar{f}}(z,\lambda ,\mu ,\gamma ). \end{aligned}$$

• (ii) If (f, g; h) is strictly generalized convex-affine on $\Omega$ at z,  then

  $$\begin{aligned} f(x)\npreceq {\bar{f}}(z,\lambda ,\mu ,\gamma ). \end{aligned}$$

Proof

Since $(z,\lambda ,\mu ,\gamma )\in C_{D}$, there exist $\lambda :=(\lambda _1,\ldots ,\lambda _m)\in \mathbb {R}^m_+{\setminus }\{0\}, \mu :=(\mu _1,\ldots ,\mu _n)\in \mathbb {R}^n_+, \gamma :=(\gamma _1,\ldots ,\gamma _l)\in \mathbb {R}^l_+, u^*_k\in \partial p_k(z), v^*_k\in \partial q_k(z), k\in K, x^*_i\in \partial g_i(z), i\in I,$ and $y^*_j\in \partial h_j(z)\cup \partial (-h_j)(z), j\in J$ such that

$$\begin{aligned}&-\left[ \sum _{k\in K}\lambda _k \left( u^*_k-\frac{p_k(z)}{q_k(z)} v^*_k\right) +\sum _{i\in I}\mu _i x^*_i+\sum _{j\in J}\gamma _j y^*_j\right] \in N(z;\Omega ),\end{aligned}$$

(4.1)

$$\begin{aligned}&\langle \mu ,g(z)\rangle +\langle \sigma ,h(z)\rangle \ge 0\; \text{ for } \text{ all } \; \sigma \in \mathbb {R}^l \text{ with } ||\sigma ||=||\gamma ||. \end{aligned}$$

(4.2)

In order to justify (i), we assume to the contrary that

$$\begin{aligned} f(x)\prec {\bar{f}}(z,\lambda ,\mu ,\gamma ). \end{aligned}$$

This means that

$$\begin{aligned} f(x)-f(z)\in -\mathrm{int\,}\mathbb {R}^m_+. \end{aligned}$$

(4.3)

By the generalized convex-affine property of (f, g; h) on $\Omega$ at z,  for such x, there exists $\nu \in N(z;\Omega )^\circ$ such that

$$\begin{aligned}&\sum _{k\in K}\lambda _k \left( \langle u^*_k,\nu \rangle -\frac{p_k(z)}{q_k(z)}\langle v_k^*,\nu \rangle \right) +\sum _{i\in I}\mu _i \langle x^*_i,\nu \rangle +\sum _{j\in J}\gamma _j \langle y^*_j,\nu \rangle \\&\quad \le \sum _{k\in K}\lambda _k \left[ p_k(x)-p_k(z)-\frac{p_k(z)}{q_k(z)}\big ( q_k(x)-q_k(z)\big )\right] +\sum _{i\in I}\mu _i \big (g_i(x)-g_i(z)\big )\\&\qquad +\sum _{j\in J}\frac{1}{\omega _j}\gamma _j \big (h_j(x)-h_j(z)\big )\\&\quad =\sum _{k\in K}\lambda _k \left( p_k(x)-\frac{p_k(z)}{q_k(z)} q_k(x)\right) +\sum _{i\in I}\mu _i \big (g_i(x)-g_i(z)\big )\\&\qquad +\sum _{j\in J}\frac{1}{\omega _j}\gamma _j \big (h_j(x)-h_j(z)\big ), \end{aligned}$$

where $\omega _j\in \{-1, 1\}, j\in J.$ Due to the definition of polar cone (2.1), it follows from (4.1) and the relation $\nu \in N(z;\Omega )^\circ$ that

$$\begin{aligned} 0\le \sum _{k\in K}\lambda _k \left( \langle u^*_k,\nu \rangle -\frac{p_k(z)}{q_k(z)}\langle v_k^*,\nu \rangle \right) +\sum _{i\in I}\mu _i \langle x^*_i,\nu \rangle +\sum _{j\in J}\gamma _j \langle y^*_j,\nu \rangle . \end{aligned}$$

Thus,

$$\begin{aligned} 0&\le \sum _{k\in K}\lambda _k \left( p_k(x)-\frac{p_k(z)}{q_k(z)} q_k(x)\right) +\sum _{i\in I}\mu _i \big (g_i(x)-g_i(z)\big )\nonumber \\&\quad +\sum _{j\in J}\sigma _j \big (h_j(x)-h_j(z)\big ), \end{aligned}$$

(4.4)

where $\sigma _j:=\frac{\gamma _j}{\omega _j}\in \mathbb {R}, j\in J.$ In addition, due to $x\in C, \sum _{i\in I}\mu _i g_i(x)\le 0$ and $\sum _{j\in J}\sigma _j h_j(x)=0$. So, (4.4) entails that

$$\begin{aligned} 0\le&\sum _{k\in K}\lambda _k \left( p_k(x)-\frac{p_k(z)}{q_k(z)} q_k(x)\right) -\sum _{i\in I}\mu _i g_i(z)-\sum _{j\in J}\sigma _jh_j(z)\\ =&\sum _{k\in K}\lambda _k \left( p_k(x)-\frac{p_k(z)}{q_k(z)} q_k(x)\right) -(\langle \mu ,g(z)\rangle +\langle \sigma ,h(z)\rangle ), \end{aligned}$$

where $\sigma :=(\sigma _1,\sigma _2,\ldots ,\sigma _l)\in \mathbb {R}^l.$ Moreover, since $||\sigma ||=||\gamma ||$, (4.2) is valid and, thus, we obtain

$$\begin{aligned} 0\le \sum _{k\in K}\lambda _k \left( p_k(x)-\frac{p_k(z)}{q_k(z)} q_k(x)\right) . \end{aligned}$$

This entails that there is $k_0\in K$ such that

$$\begin{aligned} 0\le p_{ k_0}(x)-\frac{p_{k_0}(z)}{q_{k_0}(z)} q_{k_0}(x) \end{aligned}$$

(4.5)

due to $\lambda \in \mathbb {R}^m_+{\setminus } \{0\}.$ The inequality in (4.5) is equivalent to the following one

$$\begin{aligned} f_{k_0}(z)\le f_{k_0}(x), \end{aligned}$$

which contradicts (4.3). The proof of (i) is complete.

Let us now prove (ii). Assume to the contrary that

$$\begin{aligned} f(x)\preceq {\bar{f}}(z,\lambda ,\mu ,\gamma ), \end{aligned}$$

or equivalently,

$$\begin{aligned} f(x)-f(z)\in -\mathbb {R}^m_+\backslash \{0\}. \end{aligned}$$

(4.6)

By the strictly generalized convex-affine property of (f, g; h) on $\Omega$ at z, for such x, there is $\nu \in N(z;\Omega )^\circ$ such that

$$\begin{aligned}&\sum _{k\in K}\lambda _k \left( \langle u^*_k,\nu \rangle -\frac{p_k(z)}{q_k(z)}\langle v_k^*,\nu \rangle \right) +\sum _{i\in I}\mu _i \langle x^*_i,\nu \rangle +\sum _{j\in J}\gamma _j \langle y^*_j,\nu \rangle \\&\quad < \sum _{k\in K}\lambda _k \left[ p_k(x)-p_k(z)-\frac{p_k(z)}{q_k(z)}\big ( q_k(x)-q_k(z)\big )\right] +\sum _{i\in I}\mu _i \big (g_i(x)-g_i(z)\big )\\&\quad \quad +\sum _{j\in J}\frac{1}{\omega _j}\gamma _j \big (h_j(x)-h_j(z)\big ), \end{aligned}$$

where $\omega _j\in \{-1, 1\}, j\in J.$ Proceeding similarly as in the proof of (i), we arrive at

$$\begin{aligned} 0< \sum _{k\in K}\lambda _k \left( p_k(x)-\frac{p_k(z)}{q_k(z)} q_k(x)\right) . \end{aligned}$$

This entails that there is $k_0\in K$ such that

$$\begin{aligned} 0< p_{ k_0}(x)-\frac{p_{k_0}(z)}{q_{k_0}(z)} q_{k_0}(x). \end{aligned}$$

(4.7)

Equivalently,

$$\begin{aligned} f_{k_0}(z)< f_{k_0}(x), \end{aligned}$$

which contradicts (4.6) and therefore completes the proof. $\square$

The next example asserts the importance of the generalized convex-affine property of (f, g; h) imposed in Theorem 4.1. Namely, the conclusion of the theorem may go awry if this property has been violated.

Example 4.2

Let $f: \mathbb {R}\rightarrow \mathbb {R}^2$ be defined by

$$\begin{aligned} f(x):=\left( \frac{p_1(x)}{q_1(x)},\frac{p_2(x)}{q_2(x)}\right) , \end{aligned}$$

where $p_1(x)=p_2(x):=x^3, q_1(x)=q_2(x):=x^2+1, x\in \mathbb {R},$ and let $g,h: \mathbb {R}\rightarrow \mathbb {R}$ be given by $g(x):=-|x|$ and $h(x):=x^2+x$ for $x\in \mathbb {R}.$ Consider the problem (P) with $m:=2$ and $\Omega :=(-\infty ,0]\subset \mathbb {R}$. Then $C=\{-1,0\}$ and let us select ${\bar{x}}:=-1\in C$. Now, consider the dual problem (D). By choosing ${\bar{z}}:=0\in \Omega , {\bar{\lambda }}:=(\frac{1}{2},\frac{1}{2}), {\bar{\mu }}:=1,$ and ${\bar{\gamma }}:=1,$ it holds that $({\bar{ z}},{\bar{\lambda }},{\bar{\mu }},{\bar{\gamma }})\in C_D$ and that

$$\begin{aligned} f({\bar{x}})=\left( -\frac{1}{2},-\frac{1}{2}\right) \prec (0,0)= {\bar{ f}}({\bar{z}},{\bar{\lambda }},{\bar{\mu }},{\bar{\gamma }}), \end{aligned}$$

showing that the conclusion of Theorem 4.1 fails to hold. The reason is that (f, g; h) is not generalized convex-affine on $\Omega$ at ${\bar{z}}.$

Strong duality relations between the primal problem (P) and the dual problem (D) read as follows.

Theorem 4.3

(Strong Duality) Let ${\bar{x}}\in \mathcal {S}^w(P)$ be such that the (CQ) is satisfied at this point. Then there exists $({\bar{\lambda }},{\bar{\mu }},{\bar{\gamma }})\in (\mathbb {R}^m_+\backslash \{0\})\times \mathbb {R}^n_+\times \mathbb {R}^l_+$ such that $({\bar{x}},{\bar{ \lambda }},{\bar{\mu }},{\bar{\gamma }})\in C_{D}$ and $f({\bar{x}})={\bar{f}}({\bar{x}},{\bar{ \lambda }},{\bar{\mu }},{\bar{\gamma }}).$

• (i) If in addition (f, g; h) is generalized convex-affine on $\Omega$ at any $z\in \Omega ,$ then $({\bar{x}},{\bar{\lambda }},{\bar{ \mu }},{\bar{\gamma }})\in \mathcal {S}^w(D).$

• (ii) If in addition (f, g; h) is strictly generalized convex-affine on $\Omega$ at any $z\in \Omega ,$ then $({\bar{x}},{\bar{ \lambda }},{\bar{\mu }},{\bar{\gamma }})\in \mathcal {S}(D).$

Proof

Thanks to Theorem 3.3, we find ${\bar{\lambda }}:=(\lambda _1,\ldots ,\lambda _m)\in \mathbb {R}^m_+\backslash \{0\}, {\bar{\mu }}:= (\mu _1,\ldots ,\mu _n)\in \mathbb {R}^n_+,$ and ${\bar{\gamma }}:=(\gamma _1,\ldots ,\gamma _l)\in \mathbb {R}^l_+$ such that

$$\begin{aligned} 0\in&\sum _{k\in K}\lambda _k\left( \partial p_k({\bar{ x}})-\frac{p_k({\bar{x}})}{q_k({\bar{x}})}\partial q_k({\bar{ x}})\right) +\sum _{i\in I}\mu _i\partial g_i({\bar{x}})+\sum _{j\in J}\gamma _j\big (\partial h_j({\bar{x}})\cup \partial (-h_j)({\bar{ x}})\big )\\ +\,&N({\bar{x}};\Omega ),\quad \mu _i g_i({\bar{x}})=0,\quad i\in I. \end{aligned}$$

In addition, due to ${\bar{x}}\in C, h_j({\bar{x}})=0$ for all $j\in J$, and thus, $\langle {\bar{\mu }},g({\bar{x}})\rangle +\langle \sigma ,h({\bar{ x}})\rangle =0$ for all $\sigma \in \mathbb {S}(0,||{\bar{\gamma }}||)\big ).$ So, we conclude that $({\bar{x}},{\bar{\lambda }},{\bar{\mu }},{\bar{\gamma }})\in C_{D}.$ Obviously,

$$\begin{aligned} f({\bar{x}})={\bar{f}}({\bar{x}},{\bar{\lambda }}, {\bar{\mu }},{\bar{\gamma }}). \end{aligned}$$

(i) If (f, g; h) is generalized convex-affine on $\Omega$ at any $z\in \Omega ,$ then by invoking (i) of Theorem 4.1, we obtain

$$\begin{aligned} {\bar{f}}({\bar{x}},{\bar{\lambda }}, {\bar{\mu }},{\bar{\gamma }})=f({\bar{x}})\nprec {\bar{f}}(z, \lambda , \mu ,\gamma ) \end{aligned}$$

for any $(z,\lambda , \mu ,\gamma )\in C_{D}.$ It means that $({\bar{x}},{\bar{\lambda }},{\bar{\mu }},{\bar{\gamma }})\in \mathcal {S}^w(D).$

(ii) If (f, g; h) is strictly generalized convex-affine on $\Omega$ at any $z\in \Omega ,$ then by invoking (ii) of Theorem 4.1, we assert that

$$\begin{aligned} {\bar{f}}({\bar{x}},{\bar{\lambda }}, {\bar{\mu }},{\bar{\gamma }})\npreceq {\bar{f}}(z,\lambda ,\mu ,\gamma ) \end{aligned}$$

for any $(z,\lambda , \mu ,\gamma )\in C_{D}.$ Hence, $({\bar{x}},{\bar{ \lambda }},{\bar{\mu }},{\bar{\gamma }})\in \mathcal {S}(D)$. $\square$

Remark 4.4

Observe that the (CQ) imposed in Theorem 4.3 plays an important role in establishing the strong duality results. More precisely, if ${\bar{x}}$ is a weakly efficient solution of the primal problem at which the (CQ) is not satisfied, then we might not find out a triplet $({\bar{\lambda }},{\bar{\mu }},{\bar{\gamma }})\in (\mathbb {R}^m_+\backslash \{0\})\times \mathbb {R}^n_+\times \mathbb {R}^l_+$ such that $({\bar{x}},{\bar{\lambda }},{\bar{\mu }},{\bar{\gamma }})$ belongs to the feasible/constraint set of the dual problem. In this case, of course, we do not have strong dual relations. Let us look back at Example 3.4.

We close this section by presenting converse-like duality relations between the primal problem (P) and the dual problem (D).

Theorem 4.5

(Converse Duality) Let $({\bar{x}}, {\bar{\lambda }}, {\bar{\mu }}, {\bar{\gamma }})\in C_D.$

• (i) If ${\bar{x}}\in C$ and (f, g; h) is generalized convex-affine on $\Omega$ at ${\bar{x}},$ then ${\bar{x}}\in \mathcal {S}^w(P).$

• (ii) If ${\bar{x}}\in C$ and (f, g; h) is strictly generalized convex-affine on $\Omega$ at ${\bar{x}},$ then ${\bar{x}}\in \mathcal {S}(P).$

Proof

Since $({\bar{x}},{\bar{\lambda }},{\bar{\mu }},{\bar{\gamma }})\in C_{D}$, there exist ${\bar{\lambda }}:=(\lambda _1,\ldots ,\lambda _m)\in \mathbb {R}^m_+{\setminus }\{0\}, {\bar{\mu }}:=(\mu _1,\ldots ,\mu _n)\in \mathbb {R}^n_+, {\bar{\gamma }}:=(\gamma _1,\ldots ,\gamma _l)\in \mathbb {R}^l_+, u^*_k\in \partial p_k({\bar{x}}), v^*_k\in \partial q_k({\bar{x}}), k\in K, x^*_i\in \partial g_i({\bar{x}}), i\in I,$ and $y^*_j\in \partial h_j({\bar{x}})\cup \partial (-h_j)({\bar{x}}), j\in J$ such that

$$\begin{aligned}&-\left[ \sum _{k\in K}\lambda _k \left( u^*_k-\frac{p_k({\bar{ x}})}{q_k({\bar{x}})} v^*_k\right) +\sum _{i\in I}\mu _i x^*_i+\sum _{j\in J}\gamma _j y^*_j\right] \in N({\bar{x}};\Omega ),\end{aligned}$$

(4.8)

$$\begin{aligned}&\langle {\bar{\mu }},g({\bar{x}})\rangle +\langle \sigma ,h({\bar{x}})\rangle \ge 0\; \text{ for } \text{ all } \; \sigma \in \mathbb {R}^l \text{ with } ||\sigma ||=||{\bar{\gamma }}||. \end{aligned}$$

(4.9)

It follows by (4.9) that

$$\begin{aligned} \langle {\bar{\mu }},g({\bar{x}})\rangle \ge |\langle {\bar{\gamma }},h({\bar{ x}})\rangle |\ge 0. \end{aligned}$$

(4.10)

Let ${\bar{x}}\in C$. We have $g_i({\bar{x}})\le 0$ for all $i\in I$ and thus, $\langle {\bar{\mu }},g({\bar{x}})\rangle \le 0.$ This together with (4.10) yields $\langle {\bar{\mu }},g({\bar{x}})\rangle = 0.$ Then $\mu _ig_i({\bar{x}})=0$ for all $i\in I.$ So, we assert by virtue of (4.8) that ${\bar{x}}$ satisfies condition (3.2). To finish the proof, it remains to apply Theorem 3.8. $\square$