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New class of G-Wolfe-type symmetric duality model and duality relations under \(G_{f}\)-bonvexity over arbitrary cones
Journal of Inequalities and Applications volume 2020, Article number: 30 (2020)
Abstract
This paper is devoted to theoretical aspects in nonlinear optimization, in particular, duality relations for some mathematical programming problems. In this paper, we introduce a new generalized class of second-order multiobjective symmetric G-Wolfe-type model over arbitrary cones and establish duality results under \(G_{f}\)-bonvexity/\(G_{f}\)-pseudobonvexity assumptions. We construct nontrivial numerical examples which are \(G_{f}\)-bonvex/\(G_{f}\)-pseudobonvex but neither η-bonvex/η-pseudobonvex nor η-invex/η-pseudoinvex.
1 Introduction
It is an undeniable fact that all of us are optimizers as we all make decisions for the sole purpose of maximizing our quality of life, productivity in time, and our welfare in some way or another. Since this is an ongoing struggle for creating the best possible among many inferior designs and is always the core requirement of human life, this fact yields the development of a massive number of techniques in this area, starting from the early ages of civilization until now. The efforts and lives behind this aim dedicated by many brilliant philosophers, mathematicians, scientists, and engineers have brought a high level of civilization we enjoy today. The decision process is relatively easier when there is a single criterion or object in mind. The process gets complicated when we have to make decisions in the presence of more than one criteria to judge the decisions. In such circumstances a single decision that optimizes all the criteria simultaneously may not exist. For handling such type of situations, we use multiobjective programming, also known as multiattribute optimization, which is the process of simultaneously optimizing two or more conflicting objectives subject to certain constraints. Multiobjective optimization problems can be found in various fields such as product and process design, finance, aircraft design, the oil and gas industry, automobile design, and other where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives.
Mangasarian [1] was the first who introduced the concept of second-order duality for nonlinear programming. Gulati and Gupta [2] introduced the concept of \(\eta_{1}\)-bonvexity/\(\eta _{2}\)-boncavity and derived duality results for a Wolfe-type model. The concept of G-invex function is given by Antczak [3] and derived some duality results for a constrained optimization problem. Later on, generalizing his earlier work, Antzcak [4] introduced \(G_{f}\)-invex functions for multivariate models and obtained optimality results for multiobjective programming problems. Liang et al. [5] discussed conditions for optimality and duality in a multiobjective programming problem. Bhatia and Garg [6] discussed the concept of \((V, p)\) invexity for nonsmooth vector functions and established duality results for multiobjective programs. Jayswal et al. [7] discussed multiobjective fractional programming problem involving an invex function. Stefaneseu and Ferrara [8] studied new invexities for multiobjective programming problem. Several researchers [9–21] have studied related areas.
This paper is organized as follows. In Sect. 2, we give some preliminaries and definitions used in this paper and also a nontrivial example for such type functions. In Sect. 3, we formulate second-order multiobjective symmetric G-Wolfe-type dual programs over arbitrary cones. We prove weak, strong, and converse duality theorems by using \(G_{f}\)-bonvexity/\(G_{f}\)-pseudobonvexity assumptions over arbitrary cones. Finally, we construct nontrivial numerical examples that are \(G_{f}\)-bonvex/\(G_{f}\)-pseudobonvex but neither η-bonvex/η-pseudobonvex nor η-invex/η-pseudoinvex functions.
2 Preliminaries and definitions
Let \(f = ( f_{1},f_{2},f_{3},\ldots, f_{k} ) : X\rightarrow R^{k}\) be a vector-valued differentiable function defined on a nonempty open set \(X\subseteq R^{n}\), and let \(I_{f_{i}} (X)\), \(i = 1,\ldots,k\), be the range of \(f_{i}\), that is, the image of X under \(f_{i}\). Let \(G_{f}=(G_{f_{1}},G_{f_{2}},\ldots,G_{f_{k}}):R\rightarrow R^{k}\) be a differentiable function such that every component \(G_{f_{i}}: I_{f_{i}}(X)\rightarrow R\) is strictly increasing on the range of \(I_{f_{i}}\), \(i=1,2,3,\ldots,k\).
Definition 2.1
The positive polar cone \(S^{*}\)of a cone \(S\subseteq R^{s}\) is defined by
Consider the following vector minimization problem:
where \(f=\{f_{1},f_{2},\ldots,f_{k}\}:S \rightarrow R^{k}\) and \(h=\{ h_{1},h_{2},\ldots,h_{m}\}:S \rightarrow R^{m}\) are differentiable functions on S.
Definition 2.2
\(\bar{y}\in S^{0}\) is an efficient solution of (MP) if there exists no other \(y\in S^{0}\) such that \(f_{r}(y)< f_{r}(\bar {y})\) for some \(r=1,2,3,\ldots,k\) and \(f_{i}(y)\leq f_{i}(\bar{y}) \) for all \(i=1,2,3,\ldots,k\).
Definition 2.3
If there exists a function \(\eta:S\times S \rightarrow R^{n}\) such that for all \(y\in S\),
then f is called invex at \(v\in S\) with respect to η.
Definition 2.4
If there exist \(G_{f_{i}}: I_{f_{i}}(S)\rightarrow R\) and \(\eta: S\times S \rightarrow R^{n}\) such that for all \(y\in S\),
then \(f_{i}\) is called \(G_{f_{i}}\)-pseudoinvex at \(u\in S\) with respect to η.
Definition 2.5
If there exist \(G_{f_{i}}: I_{f_{i}}(S)\rightarrow R\) and \(\eta: S\times S\rightarrow R^{n}\) such that for all \(y\in S\) and \(p\in R^{n}\),
then \(f_{i}\) is called \(G_{f_{i}}\)-bonvex at \(v\in S\) with respect to η.
Definition 2.6
If there exist functions \(G_{f}\) and \(\eta: S\times S\rightarrow R^{n}\) such that for all \(y\in S\) and \(p\in R^{n}\),
then \(f_{i}\) is called \(G_{f_{i}}\)-pseudobonvex at \(v\in S\) with respect to η.
Now, we discuss nontrivial numerical examples that are \(G_{f}\)-bonvex/\(G_{f}\)-pseudobonvex but neither η-bonvex/η-pseudobonvex nor η-invex/η-pseudoinvex functions.
Example 2.1
Let \(f:[-1,1]\rightarrow R^{4}\) be defined as
where \(f_{1}(y)=y^{10}\), \(f_{2}(y)=\arcsin y\), \(f_{3}(y)=\arctan y\), \(f_{4}(y)=\operatorname{arccot} y\), and let \(G_{f}= \{G_{f_{1}}, G_{f_{2}}, G_{f_{3}},G_{f_{4}}\}: R\rightarrow R^{4}\) be defined as
Let \(\eta:[-1,1]\times[-1,1]\rightarrow R\) be given as
To show that f is \(G_{f}\)-bonvex at \(v = 0\) with respect to η, we have to claim that
Putting the values of \(f_{i}\), \(G_{f_{i}}\), \(i=1,2,3,4\), into the last expression, after simplifying at the point \(v=0\in[-1,1]\), we clearly see from Fig. 1 that \(\pi_{i} \geq0\), \(i=1,2,3, 4\), for all \(y\in[-1,1]\). Therefore f is \(G_{f}\)-bonvex at \(v=0\in[-1,1]\) with respect to η and p.
Now, suppose
or
Therefore \(f_{3}\) is not η-bonvex at \(v=0\) with respect to p. Hence f is not η-bonvex at \(v=0\) with respect to p.
Next,
or
Therefore \(f_{3}\) is not η-invex at \(v=0\). Hence f is not η-invex at \(v=0\).
Example 2.2
Let \(f:[-2,2]\rightarrow R^{2}\) be defined as
where \(f_{1}(y)= (\frac{e^{2y}-1}{e^{y}} )\), \(f_{2}(y)= y^{3}\), and \(G_{f}= \{G_{f_{1}}, G_{f_{2}}\}: R\rightarrow R^{2}\) is defined as
Let \(\eta:[-2,2]\times[-2,2]\rightarrow R\) be given as
To show that f is \(G_{f}\)-pseudobonvex at \(v = 0\) with respect to η, we have to claim that, for \(i=1,2\),
Let
Substituting the values of η and \(f_{1}\) at the point \(v=0\), we get
Next, consider
At \(v=0\), we get \(\varphi_{1}\geq0\) for all \(y\in[-1,1]\) and p (from Fig. 3);
At the point \(v=0\), we have
Also,
At the point \(v=0\), we obtain
Hence from the expressions \(\phi_{i}\) and \(\varphi_{i}\), \(i=1, 2\), we get that f is \(G_{f}\)-pseudobonvex at \(v=0\) with respect to η.
Next, let
At the point \(v=0\), we have
Further, consider
At the point \(v=0\), we obtain
Hence \(f_{2}\) is not η-pseudobonvex at \(v=0\in[-2,2]\). Therefore \(f=(f_{1}, f_{2})\) is not η-pseudobonvex at \(v=0\in[-2,2]\).
Finally,
At the point \(v=0\), we have
Also,
At the point \(v=0\), we obtain
Hence \(f_{2}\) is not η-pseudoinvex at \(v=0\in[-2,2]\). Hence \(f=(f_{1}, f_{2})\) is not η-pseudoinvex at \(v=0\in[-2,2]\).
3 Second-order multiobjective G-Wolfe-type symmetric dual program
Consider the following pair of second-order multiobjective G-Wolfe-type dual programs over arbitrary cones.
Primal problem
(GWP)
Minimize
subject to
Dual problem
(GWD)
Maximize
subject to
where for all \(i=1,2,3,\ldots,k\),
and
- (i)
\(e_{k} =(1,1,\ldots,1)\in R^{k}\) and \(\lambda\in R^{k}\).
- (ii)
q and p are vectors in \(R^{n}\) and \(R^{m}\), respectively.
Let \(Y^{0}\) and \(Z^{0}\) be the sets of feasible solutions of (GWP) and (GWD), respectively.
Theorem 3.1
(Weak duality)
Let\((y,z,\lambda,p)\in Y^{0}\)and\((v,w,\lambda,q)\in Z^{0}\). Suppose that for all\(i=1,2,3,\ldots,k\),
- (i)
\(f_{i}(\cdot,v)\)is\(G_{f_{i}}\)-bonvex atvwith respectη,
- (ii)
\(f_{i}(x,\cdot)\)be\(G_{f_{i}}\)-boncave atywith respectη,
- (iii)
\(\eta_{1}(y,v)+u \in C_{1}\)and\(\eta_{2}(w,z) + y \in C_{2}\).
Then the following inequalities cannot hold together:
and
Proof
If possible, then suppose inequalities (5) and (6) hold. For \(\lambda> 0\), we obtain
From assumption (i) we get
Since \(\lambda> 0\), this inequality yields
From the dual constraint (3) and assumption (iii) it follows that
which implies
Using inequalities (3) and (8), we obtain
Using assumption (iv) and primal constraint (1), we get
Finally, adding inequalities (9) and (10) and using \(\lambda^{T}e_{k} =1\), we obtain
This contradicts (7). Hence the result. □
Remark 3.1
Since every \(G_{f}\)-bonvex function is \(G_{f}\)-pseudobonvex, Theorem 3.1 can also be obtained under \(G_{f}\)-pseudobonvexity assumptions.
Remark 3.2
A vector space V over field K, the span of a set S, may be defined as the set of all finite linear combinations of elements (vectors) of S:
Theorem 3.2
(Strong duality)
Let\((\bar{y},\bar{z},\bar {\lambda},\bar{p})\)be an efficient solution of (GWP); fix\(\lambda= \bar{\lambda}\)in (GWD) such that
- (i)
for all\(i=1,2,3,\ldots,k\), \([G_{f_{i}}^{\prime\prime }(f_{i}(\bar{y},\bar{z}))\nabla_{z}f_{i}(\bar{y},\bar{z})(\nabla _{z}f_{i}(\bar{y},\bar{z}))^{T}+ G_{f_{i}}^{\prime}(f_{i}(\bar{y},\bar {z}))\nabla_{zz}f_{i}(\bar{y},\bar{z})]\)is nonsingular,
- (ii)
\(\sum_{i=1}^{k}\bar{\lambda}_{i}\nabla_{z} (\{ G_{f_{i}}^{\prime\prime}(f_{i}(\bar{y},\bar{z})) \nabla_{z}f_{i}(\bar {y},\bar{z})(\nabla_{z}f_{i}(\bar{y},\bar{z}))^{T}+ G_{f_{i}}^{\prime }(f_{i}(\bar{y},\bar{z}))\nabla_{zz}f_{i}(\bar{y},\bar{z})\}\bar{p} )\bar{p} \notin \operatorname{span} \{G_{f_{1}}^{\prime}(f_{1}(\bar {y},\bar{z}))\nabla_{z}f_{1}(\bar{z},\bar{x}),\ldots, G_{f_{k}}^{\prime }(f_{k}(\bar{y},\bar{z}))\nabla_{z}f_{k}(\bar{y},\bar{z}) \} \setminus\{0\}\),
- (iii)
the vectors\(\{G_{f_{1}}^{\prime}(f_{1}(\bar{y},\bar {z}))\nabla_{z}f_{1}(\bar{z},\bar{x}),G_{f_{2}}^{\prime}(f_{2}(\bar {y},\bar{z}))\nabla_{z}f_{2}(\bar{y},\bar{z}),\ldots,G_{f_{k}}^{\prime }(f_{k}(\bar{y},\bar{z}))\nabla_{z}f_{k}(\bar{y},\bar{z}) \}\)are linearly independent,
- (iv)
\(\sum_{i=1}^{k}\bar{\lambda}_{i}\nabla_{y} (\{ G_{f_{i}}^{\prime\prime}(f_{i}(\bar{y},\bar{z})) \nabla_{z}f_{i}(\bar {y},\bar{z})(\nabla_{z}f_{i}(\bar{y},\bar{z}))^{T} + G_{f_{i}}^{\prime }(f_{i}(\bar{y},\bar{z}))\nabla_{zz}f_{i}(\bar{y},\bar{z})\}\bar{p} )\bar{p} =0\)implies that\(\bar{p}=0\).
Then for\(\bar{q}=0\), we have\((\bar{v},\bar{w},\bar{\lambda},\bar {p}=0)\in Z^{0}\)and\(R(\bar{y},\bar{z},\bar{\lambda},\bar{q})=S(\bar {y},\bar{z},\bar{\lambda},\bar{q})\). Also, from Theorem 3.1it follows that\((\bar{v},\bar{w},\bar{\lambda},\bar{p}=0)\)is an efficient solution for (GWD).
Proof
By the Fritz–John necessary conditions [22] there exist \(\alpha\in R^{k}\), \(\beta\in R^{m}\), and \(\eta\in R\) such that
Equation (14) can be rewritten as
By assumption (i), since \(\bar{\lambda}_{i} > 0\) for \(i=1,2,3,\ldots,k\), (18) gives
If \(\alpha=0\), then (19) implies that \(\beta=0\). Further, equation (18) gives \(\eta=0\). Consequently, \((\alpha,\beta,\eta )=0\), which contradicts (17). Hence \(\alpha\neq0\), or \(\alpha^{T} e_{k} > 0\).
Using (19) and \(\alpha^{T}e_{k}> 0\) in (12), we get
It follows from assumption (ii) that
Hence by assumption (iv) we get \(\bar{p}=0\), and therefore inequality (19) implies
Now, using \(\bar{p}=0\) and (20), we obtain
Assumption (iii) yields
Using \(\alpha^{T}e_{k}> 0\) and (21)–(23) in (11), we get
Let \(y\in C_{1}\). Then, \(y +\bar{y}\in C_{1}\), and it follows that
Therefore
Also, from (22) we have
Hence \((\bar{v},\bar{w},\bar{\lambda},\bar{p}=0)\) satisfies the dual constraints and \(Z^{0}\).
Now, letting \(y=0\) and \(y=2\bar{y}\) in (24), we get
Using (28) and \(\bar{q}=\bar{p}=0\) completes the proof. □
Theorem 3.3
(Converse duality)
Let\((\bar{v},\bar{w},\bar {\lambda},\bar{q})\)be an efficient solution of (GWD). Fix\(\lambda= \bar{\lambda}\)in (GWP) such that
- (i)
for all\(i=1,2,3,\ldots,k\), \([G_{f_{i}}^{\prime\prime }(f_{i}(\bar{v},\bar{w}))\nabla_{z}f_{i}(\bar{v},\bar{w})(\nabla _{z}f_{i}(\bar{v},\bar{w}))^{T} + G_{f_{i}}^{\prime}(f_{i}(\bar{v},\bar {w}))\nabla_{zz} f_{i}(\bar{v},\bar{w})]\)is nonsingular,
- (ii)
\(\sum_{i=1}^{k}\bar{\lambda}_{i}\nabla_{z} (\{ G_{f_{i}}^{\prime\prime}(f_{i}(\bar{v},\bar{w}))\nabla_{z}f_{i}(\bar {v},\bar{w})(\nabla_{z}f_{i}(\bar{v},\bar{w}))^{T} + G_{f_{i}}^{\prime }(f_{i}(\bar{v},\bar{w}))\nabla_{zz}f_{i}(\bar{v},\bar{w})\} \bar{q} )\bar{q} \notin \operatorname{span} \{G_{f_{1}}^{\prime }(f_{1}(\bar{v},\bar{w}))\nabla_{z}f_{1}(\bar{v},\bar{w}),\ldots, G_{f_{k}}^{\prime}(f_{k}(\bar{u},\bar{v}))\nabla_{z}f_{k}(\bar{u},\bar {v}) \}\setminus\{0\}\),
- (iii)
the vectors\(\{G_{f_{1}}^{\prime}(f_{1}(\bar{v},\bar {w}))\nabla_{z}f_{1}(\bar{v},\bar{w}),G_{f_{2}}^{\prime}(f_{2}(\bar {v},\bar{w}))\nabla_{z}f_{2}(\bar{v},\bar{w}),\ldots,G_{f_{k}}^{\prime }(f_{k}(\bar{v},\bar{w}))\nabla_{z}f_{k}(\bar{v},\bar{w}) \}\)are linearly independent,
- (iv)
\(\sum_{i=1}^{k}\bar{\lambda}_{i}\nabla_{z} (\{ G_{f_{i}}^{\prime\prime}(f_{i}(\bar{v},\bar{w}))\nabla_{z}f_{i}(\bar {v},\bar{w})(\nabla_{z}f_{i}(\bar{v},\bar{w}))^{T} + G_{f_{i}}^{\prime }(f_{i}(\bar{v},\bar{w}))\nabla_{zz}f_{i}(\bar{v},\bar{w})\}\bar{q} )\bar{q}=0\Rightarrow \bar{q}=0\).
Then, taking\(\bar{p}=0\), we have that\((\bar{v},\bar{w},\bar{\lambda },\bar{p}=0)\in Y^{0}\)and\(R(\bar{v},\bar{w},\bar{\lambda},\bar{p})= S(\bar{u},\bar{v},\bar{\lambda},\bar{p})\). Also, by Theorem3.1\((\bar{v},\bar{w},\bar{\lambda},\bar{p}=0)\)is an efficient solution for (GWP).
Proof
Proof follows the lines of Theorem 3.2. □
4 Concluding remarks
In this paper, we have formulated a second-order symmetric G-Wolfe-type dual problem for a nonlinear multiobjective optimization problem with cone constraints. A number of duality relations are further established under \(G_{f}\)-bonvexity/\(G_{f}\)-pseudobonvexity assumptions on the function f. We have discussed various numerical examples to show the existence of \(G_{f}\)-bonvex/\(G_{f}\)-pseudobonvex functions. The question arises whether the duality results developed in this paper hold for G-Wolfe- or mixed-type higher-order multiobjective optimization problems. This may be the future direction for the researchers working in this area.
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Acknowledgements
The authors are thankful to the anonymous referees and editor for their valuable suggestions, which have substantially improved the presentation of the paper.
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The authors extend their appreciation to the “Deanship of Scientific Research” at King Khalid University for funding this work through research groups program under grant R.G.P.1/152/40.
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Dubey, R., Kumar, A., Ali, R. et al. New class of G-Wolfe-type symmetric duality model and duality relations under \(G_{f}\)-bonvexity over arbitrary cones. J Inequal Appl 2020, 30 (2020). https://doi.org/10.1186/s13660-019-2279-0
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DOI: https://doi.org/10.1186/s13660-019-2279-0