Generalized Second-Order Mixed Symmetric Duality in Nondifferentiable Mathematical Programming

Abstract

This paper is concerned with a pair of second-order mixed symmetric dual programs involving nondifferentiable functions. Weak, strong, and converse duality theorems are proved for aforementioned pair using the notion of second-order 𝐹-convexity/pseudoconvexity assumptions.

1. Introduction

Duality is a fruitful theory in mathematical programming and is useful both theoretically and practically. Duality as used in our daily life means the sort of harmony of two opposite or complementary parts through which they integrate into a whole. Symmetry is bound up with duality and, in particular, is significant in mathematics. The problem of optimizing a numerical function of one or more variables subject to constraints on the variables is called the mathematical programming, or constrained optimization, problem. When either the objective function or one or more of the constraints are nonlinear, the programming problem is called a nonlinear programming problem, a discipline playing an increasingly imperative role in such diverse fields as operations research and management science, engineering, economics, system analysis, and computer science.

Dantzig et al. [1], Mond [2], and Bazaraa and Goode [3] studied symmetric duality in nonlinear programming. Later, Chandra and Husain [4] formulated a pair of Wolfe-type nondifferentiable symmetric dual programs and proved duality results under convexity/concavity assumptions. Subsequently, Chandra et al. [5] weakened these assumptions to pseudoconvexity/pseudoconcavity. Mond and Schechter [6] presented two symmetric dual pairs involving nondifferentiable functions. Kumar and Bhatia [7] discussed multiobjective symmetric duality by using a nonlinear vector-valued function of two variables corresponding to various objectives.

Mangasarian [8] presented a dual problem associated with a primal nonlinear programming problem that involves second derivatives of the function constituting the primal problem. The study of second-order duality is significant, as it can provide a lower bound to the infimum of a primal optimization problem when it is difficult to find a feasible solution for the first-order dual. Bector and Chandra [9] achieved duality results for a pair of Mond-Weir-type second-order symmetric dual nonlinear programs. Hou and Yang [10] formulated a pair of second-order symmetric dual nondifferentiable programs and established duality theorems under second-order 𝐹-pseudoconvexity assumptions.

Chandra et al. [11] and Yang et al. [12] discussed a mixed symmetric dual formulation for a nonlinear programming problem and for a class of nondifferentiable nonlinear programming problems, respectively. Later on, Ahmad [13] formulated mixed type symmetric dual in multiobjective programming problems ignoring nonnegativity restrictions of Bector et al. [14].

In this paper, a pair of second-order mixed symmetric dual programs is presented for a class of nondifferentiable nonlinear programming problems. Weak, strong, and converse duality theorems are proved using the notion of second-order 𝐹-convexity/pseudoconvexity assumptions. These results generalize the known work in [6, 10–12, 15–17].

2. Preliminaries

In this section, we presented some of the basic definitions used in the paper.

Definition 2.1. Let 𝐶 be a compact convex set in 𝑅^𝑛. The support function of 𝐶 is defined by 𝑆(𝑥∣𝐶)=max𝑥^𝑇𝑦∶𝑦∈𝐶. (2.1) A support function, being convex and everywhere finite, has a subdifferential, that is, there exists 𝑧∈𝑅^𝑛 such that 𝑆𝑦∣𝐶≥𝑆(𝑥∣𝐶)+𝑧^𝑇𝑦−𝑥,∀𝑦∈𝐶. (2.2) The subdifferential of 𝑆(𝑥∣𝐶) is given by 𝜕𝑆(𝑥∣𝐶)=𝑧∈𝐶∶𝑧^𝑇𝑥=𝑆(𝑥∣𝐶). (2.3) For any set 𝑆⊂𝑅^𝑛, the normal cone to 𝑆 at a point 𝑥∈𝑆 is defined by 𝑁_𝑆(𝑥)=𝑦∈𝑅^𝑛∶𝑦^𝑇(𝑧−𝑥)≤0,∀𝑧∈𝑆. (2.4) It can be easily seen that for a compact convex set 𝐶, 𝑦 is in 𝑁_𝐶(𝑥) if and only if 𝑆(𝑦∣𝐶)=𝑥^𝑇𝑦, or equivalently, 𝑥 is in 𝜕𝑆(𝑦∣𝐶).

Definition 2.2. A functional 𝐹∶𝑋×𝑋×𝑅^𝑛↦𝑅 (where 𝑋⊆𝑅^𝑛) is sublinear with respect to the third variable if for all (𝑥,𝑢)∈𝑋×𝑋,(i)𝐹(𝑥,𝑢;𝑎₁+𝑎₂)≤𝐹(𝑥,𝑢;𝑎₁)+𝐹(𝑥,𝑢;𝑎₂) for all 𝑎₁,𝑎₂∈𝑅^𝑛, (ii)𝐹(𝑥,𝑢;𝛼𝑎)=𝛼𝐹(𝑥,𝑢;𝑎), for all 𝛼∈𝑅₊ and for all 𝑎∈𝑅^𝑛. Let 𝜓∶𝑋↦𝑅 be a real-valued twice differentiable function.

Definition 2.3. 𝜓 is said to be second-order 𝐹-convex at 𝑢∈𝑋 with respect to 𝑞∈𝑅^𝑛, if for all 𝑥∈𝑋, 𝜓(𝑥)(𝑢)+1−𝜓2𝑞^𝑇∇_𝑥𝑥𝜓(𝑢)𝑞≥𝐹𝑥,𝑢;∇_𝑥𝜓(𝑢)+∇_𝑥𝑥𝜓(𝑢)𝑞. (2.5)

Definition 2.4. 𝜓 is said to be second-order 𝐹-pseudoconvex at 𝑢∈𝑋 with respect to 𝑞∈𝑅^𝑛, if for all 𝑥∈𝑋, 𝐹𝑥,𝑢;∇_x𝜓(𝑢)+∇_𝑥𝑥𝜓(𝑢)𝑞(𝑥)(𝑢)−1≥0⟹𝜓≥𝜓2𝑞^𝑇∇_𝑥𝑥𝜓(𝑢)𝑞. (2.6)𝜓 is second-order 𝐹-concave/pseudoconcave at 𝑢∈𝑋 with respect to 𝑞∈𝑅^𝑛 if −𝜓 is second-order 𝐹-convex/pseudoconvex at 𝑢∈𝑋 with respect to 𝑞∈𝑅^𝑛.

3. Second-Order Mixed Nondifferentiable Symmetric Dual Programs

For 𝑁={1,2,…,𝑛} and 𝑀={1,2,…,𝑚}, let 𝐽₁⊆𝑁, 𝐾₁⊆𝑀, 𝐽₂=𝑁⧵𝐽₁, and 𝐾₂=𝑀⧵𝐾₁. Let |𝐽₁| denote the number of elements in 𝐽₁. The other symbols |𝐽₂|, |𝐾₁| and |𝐾₂| are defined similarly. Let 𝑥¹∈𝑅^|𝐽1|, 𝑥²∈𝑅^|𝐽2|. Then, any 𝑥∈𝑅^𝑛 can be written as (𝑥¹,𝑥²). Similarly, for 𝑦¹∈𝑅^|𝐾1|, 𝑦²∈𝑅^|𝐾2|, 𝑦∈𝑅^𝑚 can be written as (𝑦¹,𝑦²). It may be noted here that if 𝐽₁=∅, then |𝐽₁|=0, 𝐽₂=𝑁, and therefore |𝐽₂|=𝑛. In this case, 𝑅^|𝐽1|,𝑅^|𝐽2| and 𝑅^|𝐽1|×𝑅^|𝐾1| will be zero-dimensional, 𝑛-dimensional and |𝐾₁|-dimensional Euclidean spaces, respectively. The other situations are 𝐽₂=∅, 𝐾₁=∅ or 𝐾₂=∅.

Now we formulate the following pair of mixed nondifferentiable second-order symmetric dual programs and discuss their duality results.

Primal problem (SMNP)
minimize 𝐺𝑥¹,𝑦¹,𝑥²,𝑦²,𝑧²𝑥,𝑝,𝑟=𝑓¹,𝑦¹+𝑆𝑥¹∣𝐶₁+𝑔𝑥²,𝑦²+𝑆𝑥²∣𝐶₂−𝑦^2𝑇𝑧²−𝑦^1𝑇∇_𝑦1𝑓𝑥¹,𝑦¹+∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝−12𝑝^𝑇∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝−12𝑟^𝑇∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝑟, (3.1) subject to ∇_𝑦1𝑓𝑥¹,𝑦¹−𝑧¹+∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹∇𝑝≤0,_𝑦2𝑔𝑥²,𝑦²−𝑧²+∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝑦𝑟≤0,_2𝑇∇_𝑦2𝑔𝑥²,𝑦²−𝑧²+∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝑟𝑧≥0,₁∈𝐷₁,𝑧²∈𝐷₂. (3.2)(3.3)(3.4)(3.5)

Dual problem (SMND)
maximize 𝐻𝑢¹,𝑣¹,𝑢²,𝑣²,𝑤²𝑢,𝑞,𝑠=𝑓¹,𝑣¹−𝑆𝑣¹∣𝐷₁+𝑔𝑢²,𝑣²−𝑆𝑣²∣𝐷₂+𝑢^2𝑇𝑤²−𝑢^1𝑇∇_𝑥1𝑓𝑢¹,𝑣¹+∇_𝑥1𝑥1𝑓𝑢¹,𝑣¹𝑞−12𝑞^𝑇∇_𝑥1𝑥1𝑓𝑢¹,𝑣¹𝑞−12𝑠^𝑇∇_𝑥2𝑥2𝑔𝑢²,𝑣²𝑠, (3.6) subject to ∇_𝑥1𝑓𝑢¹,𝑣¹+𝑤¹+∇_𝑥1𝑥1𝑓𝑢¹,𝑣¹∇𝑞≥0,_𝑥2𝑔𝑢²,𝑣²+𝑤²+∇_𝑥2𝑥2𝑔𝑢²,𝑣²𝑢𝑠≥0,_2𝑇∇_𝑥2𝑔𝑢²,𝑣²+𝑤²+∇_𝑥2𝑥2𝑔𝑢²,𝑣²𝑠𝑤≤0,₁∈𝐶₁,𝑤²∈𝐶₂, (3.7)(3.8)(3.9)(3.10) where (i)𝑓∶𝑅^|𝐽1|×𝑅^|𝐾1|→𝑅 and 𝑔∶𝑅^|𝐽2|×𝑅^|𝐾2|→𝑅 are differentiable functions, (ii)𝐶₁, 𝐶₂, 𝐷₁ and 𝐷₂ are compact convex sets in 𝑅^|𝐽1|, 𝑅^|𝐽2|, 𝑅^|𝐾1| and 𝑅^|𝐾2|, respectively, (iii)𝑝∈𝑅^|𝐾1|, 𝑟∈𝑅^|𝐾2|, 𝑞∈𝑅^|𝐽1| and 𝑠∈𝑅^|𝐽2|.

Theorem 3.1 (Weak duality). Let (𝑥¹,𝑦¹,𝑥²,𝑦²,𝑧¹,𝑧²,𝑝,𝑟) be feasible for (SMNP) and (𝑢¹,𝑣¹,𝑢²,𝑣²,𝑤¹,𝑤²,𝑞,𝑠) be feasible for (SMND). Let the sublinear functionals 𝐹₁∶𝑅^|𝐽1|×𝑅^|𝐽1|×𝑅^|𝐽1|↦𝑅, 𝐹₂∶𝑅^|𝐾1|×𝑅^|𝐾1|×𝑅^|𝐾1|↦𝑅, 𝐺₁∶𝑅^|𝐽2|×𝑅^|𝐽2|×𝑅^|𝐽2|↦𝑅 and 𝐺₂∶𝑅^|𝐾2|×𝑅^|𝐾2|×𝑅^|𝐾2|↦𝑅 satisfy the following conditions: 𝐹₁𝑥¹,𝑢¹;𝑎¹+𝑎^1𝑇𝑢¹≥0,∀𝑎¹∈𝑅^|𝐽1|₊,𝐹₂𝑣¹,y¹;𝑎²+𝑎^2𝑇𝑦¹≥0,∀𝑎²∈𝑅^|𝐾1|₊𝐺₁,𝑥²,𝑢²;𝑏¹+𝑏^1𝑇𝑢²≥0,∀𝑏¹∈𝑅^|𝐽2|₊𝐺₂,𝑣²,𝑦²;𝑏²+𝑏^2𝑇𝑦²≥0,∀𝑏²∈𝑅^|𝐾2|₊. (A)(B)(C)(D) Suppose that (i)𝑓(⋅,𝑣¹)+(⋅)^𝑇𝑤¹ is second-order 𝐹₁-convex at 𝑢¹, and 𝑓(𝑥¹,⋅)−(⋅)^𝑇𝑧¹ is second-order 𝐹₂-concave at 𝑦¹, (ii)𝑔(⋅,𝑣²)+(⋅)^𝑇𝑤² is second-order 𝐺₁-pseudoconvex at 𝑢², and 𝑔(𝑥²,⋅)−(⋅)^𝑇𝑧² is second-order 𝐺₂-pseudoconcave at 𝑦². Then, 𝐺𝑥¹,𝑦¹,𝑥²,𝑦²,𝑧²,𝑝,𝑟≥𝐻𝑢¹,𝑣¹,𝑢²,𝑣²,𝑤²,𝑞,𝑠. (3.11)

Proof. By the second-order 𝐹₁-convexity of 𝑓(⋅,𝑣¹)+(⋅)^𝑇𝑤¹ at 𝑢¹ and the second-order 𝐹₂-concavity of 𝑓(𝑥¹,⋅)−(⋅)^𝑇𝑧¹ at 𝑦¹, we have 𝑓𝑥¹,𝑣¹+𝑥^1𝑇𝑤¹−𝑓𝑢¹,𝑣¹−𝑢^1𝑇𝑤¹+12𝑞^𝑇∇_𝑥1𝑥1𝑓𝑢¹,𝑣¹𝑞≥𝐹₁𝑥¹,𝑢¹;∇_𝑥1𝑓𝑢¹,𝑣¹+𝑤¹+∇_𝑥1𝑥1𝑓𝑢¹,𝑣¹𝑞,𝑓𝑥¹,𝑦¹−𝑦^1𝑇𝑧₁−𝑓𝑥¹,𝑣¹+𝑣^1𝑇𝑧₁−12𝑝^𝑇∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝≥𝐹₂𝑣¹,𝑦¹;−∇_𝑦1𝑓𝑥¹,𝑦¹−𝑧¹+∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝. (3.12)(3.13) Since (𝑥¹,𝑦¹,𝑥²,𝑦²,𝑧¹,𝑧²,𝑝,𝑟) is feasible for primal problem (SMNP) and (𝑢¹,𝑣¹,𝑢²,𝑣²,𝑤¹,𝑤²,𝑞,𝑠) is feasible for dual problem (SMND), by the dual constraint (3.7), the vector 𝑎¹=∇_𝑥1𝑓(𝑢¹,𝑣¹)+𝑤¹+∇_𝑥1𝑥1𝑓(𝑢¹,𝑣¹)𝑞∈𝑅^|𝐽1|₊, and so from the hypothesis (A), we obtain 𝐹₁𝑥¹,𝑢¹;𝑎¹+𝑎^1𝑇𝑢¹≥0. (3.14) Similarly, 𝐹₂𝑣¹,𝑦¹;𝑎²+𝑎^2𝑇𝑦¹≥0, (3.15) for the vector 𝑎²=−[∇_𝑦1𝑓(𝑥¹,𝑦¹)−𝑧¹+∇_𝑦1𝑦1𝑓(𝑥¹,𝑦¹)𝑝]∈𝑅^|𝐾1|₊.
Using (3.14) in (3.12) and (3.15) in (3.13), we have 𝑓𝑥¹,𝑣¹+𝑥^1𝑇𝑤¹−𝑓𝑢¹,𝑣¹−𝑢^1𝑇𝑤¹+12𝑞^𝑇∇_𝑥1𝑥1𝑓𝑢¹,𝑣¹𝑞≥−𝑢^1𝑇𝑎¹,𝑓𝑥¹,𝑦¹−𝑦^1𝑇𝑧₁−𝑓𝑥¹,𝑣¹+𝑣^1𝑇𝑧₁−12𝑝^𝑇∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝≥−𝑦^1𝑇𝑎². (3.16) Adding the above two inequalities, we obtain 𝑓𝑥¹,𝑦¹+𝑥^1𝑇𝑤¹−𝑦^1𝑇𝑧₁+𝑦^1𝑇𝑎²−12𝑝^𝑇∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝≥𝑓𝑢¹,𝑣¹+𝑢^1𝑇𝑤¹−𝑣^1𝑇𝑧₁−𝑢^1𝑇𝑎¹−12𝑞^𝑇∇_𝑥1𝑥1𝑓𝑢¹,𝑣¹𝑞. (3.17) Substituting the values of 𝑎¹ and 𝑎² in (3.17), we get 𝑓𝑥¹,𝑦¹+𝑥^1𝑇𝑤¹−𝑦^1𝑇∇_𝑦1𝑓𝑥¹,𝑦¹+∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝−12𝑝^𝑇∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝≥𝑓𝑢¹,𝑣¹−𝑣^1𝑇𝑧₁−𝑢^1𝑇∇_𝑥1𝑓𝑢¹,𝑣¹+∇_𝑥1𝑥1𝑓𝑢¹,𝑣¹𝑞−12𝑞^𝑇∇_𝑥1𝑥1𝑓𝑢¹,𝑣¹𝑞. (3.18) Using (𝑥¹)^𝑇𝑤¹≤𝑆(𝑥¹∣𝐶₁) and (𝑣¹)^𝑇𝑧¹≤𝑆(𝑣¹∣𝐷₁), we have 𝑓𝑥¹,𝑦¹+𝑆𝑥¹∣𝐶₁−𝑦^1𝑇∇_𝑦1𝑓𝑥¹,𝑦¹+∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝−12𝑝^𝑇∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝≥𝑓𝑢¹,𝑣¹−𝑆𝑣¹∣𝐷₁−𝑢^1𝑇∇_𝑥1𝑓𝑢¹,𝑣¹+∇_𝑥1𝑥1𝑓𝑢¹,𝑣¹𝑞−12𝑞^𝑇∇_𝑥1𝑥1𝑓𝑢¹,𝑣¹𝑞. (3.19) By hypothesis (C) and the dual constraint (3.8), we obtain 𝐺₁𝑥²,𝑢²;∇_𝑥2𝑔𝑢²,𝑣²+𝑤²+∇_𝑥2𝑥2𝑔𝑢²,𝑣²𝑠≥−𝑢^2𝑇∇_𝑥2𝑔𝑢²,𝑣²+𝑤²+∇_𝑥2𝑥2𝑔𝑢²,𝑣²𝑠, (3.20) which on using the dual constraint (3.9) yields 𝐺₁𝑥²,𝑢²;∇_𝑥2𝑔𝑢²,𝑣²+𝑤²+∇_𝑥2𝑥2g𝑢²,𝑣²𝑠≥0. (3.21) Since 𝑔(⋅,𝑣²)+(⋅)^𝑇𝑤² is second-order 𝐺₁-pseudoconvex at 𝑢², we have 𝑔𝑥²,𝑣²+𝑥^2𝑇𝑤²≥𝑔𝑢²,𝑣²+𝑢^2𝑇𝑤²−12𝑠^𝑇∇_𝑥2𝑥2𝑔𝑢²,𝑣²𝑠. (3.22) Similarly, from (3.3) and (3.4) and hypothesis (D) along with second-order 𝐺₂-pseudoconcavity of 𝑔(𝑥²,⋅)−(⋅)^𝑇𝑧² at 𝑦², we get 𝑔𝑥²,𝑦²−𝑦^2𝑇𝑧²≥𝑔𝑥²,𝑣²−𝑣^2𝑇𝑧²+12𝑟^𝑇∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝑟. (3.23) Adding (3.22) and (3.23), we obtain 𝑔𝑥²,𝑦²+𝑥^2𝑇𝑤²−𝑦^2𝑇𝑧²−12𝑟^𝑇∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝑟≥𝑔𝑢²,𝑣²+𝑢^2𝑇𝑤²−𝑣^2𝑇𝑧²−12𝑠^𝑇∇_𝑥2𝑥2𝑔𝑢²,𝑣²𝑠. (3.24) Using (𝑥²)^𝑇𝑤²≤𝑆(𝑥²|𝐶₂) and (𝑣²)^𝑇𝑧²≤𝑆(𝑣²|𝐷₂), we have 𝑔𝑥²,𝑦²+𝑆𝑥²∣𝐶₂−𝑦^2𝑇𝑧²−12𝑟^𝑇∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝑟≥𝑔𝑢²,𝑣²+𝑢^2𝑇𝑤²−𝑆𝑣²∣𝐷₂−12𝑠^𝑇∇_𝑥2𝑥2𝑔𝑢²,𝑣²𝑠. (3.25) Inequalities (3.19) and (3.25) together yield 𝑓𝑥¹,𝑦¹+𝑆𝑥¹∣𝐶₁+𝑔𝑥²,𝑦²+𝑆𝑥²∣𝐶₂−𝑦^2𝑇𝑧²−𝑦^1𝑇∇_𝑦1𝑓𝑥¹,𝑦¹+∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝−12𝑝^𝑇∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹1𝑝−2𝑟^𝑇∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝑟≥𝑓𝑢¹,𝑣¹−𝑆𝑣¹∣𝐷₁+𝑔𝑢²,𝑣²−𝑆𝑣²∣𝐷₂+𝑢^2𝑇𝑤²−𝑢^1𝑇∇_𝑥1𝑓𝑢¹,𝑣¹+∇_𝑥1𝑥1𝑓𝑢¹,𝑣¹𝑞−12𝑞^𝑇∇_𝑥1𝑥1𝑓𝑢¹,𝑣¹𝑞−12𝑠^𝑇∇_𝑥2𝑥2𝑔𝑢²,𝑣²𝑠, (3.26) that is,𝐺(𝑥¹,𝑦¹,𝑥²,𝑦²,𝑧²,𝑝,𝑟)≥𝐻(𝑢¹,𝑣¹,𝑢²,𝑣²,𝑤²,𝑞,𝑠).

Theorem 3.2 (Strong duality). Let 𝑓∶𝑅^|𝐽1|×𝑅^|𝐾1|→𝑅 and 𝑔∶𝑅^|𝐽2|×𝑅^|𝐾2|→𝑅 be differentiable functions, and let (𝑥¹,𝑦¹,𝑥²,𝑦²,𝑧¹,𝑧²,𝑝,𝑟) be a local optimal solution of (SMNP). Suppose that (i)the matrix ∇_𝑦1𝑦1𝑓(𝑥¹,𝑦¹) is non singular, (ii)∇_𝑦2𝑦2𝑔(𝑥²,𝑦²) is positive definite, 𝑟^𝑇(∇_𝑦2𝑔(𝑥²,𝑦²)−𝑧²)≥0 or ∇_𝑦2𝑦2𝑔(𝑥²,𝑦²) is negative definite, and 𝑟^𝑇(∇_𝑦2𝑔(𝑥²,𝑦²)−𝑧²)≤0, (iii)∇_𝑦2𝑔(𝑥²,𝑦²)−𝑧²+∇_𝑦2𝑦2𝑔(𝑥²,𝑦²)𝑟≠0, (iv)one of the matrices (𝜕/𝜕𝑦¹_𝑖)(∇_𝑦1𝑦1𝑓(𝑥¹,𝑦¹)),  𝑖=1,2,…,|𝐾₁|, is positive or negative definite. Then, there exist 𝑤¹∈𝐶₁ and 𝑤²∈𝐶₂ such that (𝑥¹,𝑦¹,𝑥²,𝑦²,𝑤¹,𝑤²,𝑞=0,𝑠=0) is feasible for (SMND), and the objective function values of (SMNP) and (SMND) are equal. Furthermore, if the assumptions of weak duality (Theorem 3.1) are satisfied for all feasible solutions of (SMNP) and (SMND), then (𝑥¹,𝑦¹,𝑥²,𝑦²,z¹,𝑧²,𝑝,𝑟) and (𝑥¹,𝑦¹,𝑥²,𝑦²,𝑤¹,𝑤²,𝑞,𝑠) are global optimal solutions for (SMNP) and (SMND), respectively.

Proof. Since (𝑥¹,𝑦¹,𝑥²,𝑦²,𝑧¹,𝑧²,𝑝,𝑟) is a local optimal solution of (SMNP), there exist 𝛼∈𝑅, 𝛽∈𝑅^|𝐾1|, 𝛾∈𝑅^|𝐾2|, 𝛿∈𝑅, 𝜂₁∈𝑅^|𝐽1|, and 𝜂₂∈𝑅^|𝐽2| such that the following by Fritz John optimality conditions [18] are satisfied at (𝑥¹,𝑦¹,𝑥²,𝑦²,𝑧¹,𝑧²,𝑝,𝑟)𝛼^𝑇∇_𝑥1𝑓𝑥¹,𝑦¹+𝜂₁+∇_𝑦1𝑥1𝑓𝑥¹,𝑦¹𝛽−𝛼𝑦¹+∇_𝑥1∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝𝛽−𝛼𝑦¹+12𝑝𝛼=0,_𝑇∇_𝑥2𝑔𝑥²,𝑦²+𝜂₂+∇_𝑦2𝑥2𝑔𝑥²,𝑦²𝛾−𝛿𝑦²+∇_𝑥2∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝑟𝛾−𝛿𝑦²−12𝛼𝑟∇=0,_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝛽−𝛼𝑦¹+𝑝+∇_𝑦1∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝𝛽−𝛼𝑦¹+12𝑝∇=0,_𝑦2𝑔𝑥²,𝑦²−𝑧²[𝛼−𝛿]+∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝛾−𝛿𝑦²+𝑟+∇_𝑦2∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝑟𝛾−𝛿𝑦²−12𝛼𝑟∇=0,_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝛽−𝛼𝑦¹+𝑝∇=0,_𝑦2𝑦2𝑔𝑥²,𝑦²𝛾−𝛿𝑦²−𝛼𝑟𝛽=0,_𝑇∇_𝑦1𝑓𝑥¹,𝑦¹−𝑧¹+∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝𝛾=0,_𝑇∇_𝑦2𝑔𝑥²,𝑦²−𝑧²+∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝑟𝛿=0,𝑦^2𝑇∇_𝑦2𝑔𝑥²,𝑦²−𝑧²+∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝑟=0,𝛽∈𝑁_𝐷1𝑧¹(),𝛼−𝛿𝑦²+𝛾∈𝑁_𝐷2𝑧²𝜂₁,∈𝐶₁,𝜂^𝑇₁𝑥¹=𝑆𝑥¹∣𝐶₁𝜂₂,∈𝐶₂,𝜂^𝑇₂𝑥²=𝑆𝑥²∣𝐶₂,𝛼,𝛽,𝛾,𝛿≥0,𝛼,𝛽,𝛾,𝛿≠0. (3.27)(3.28)(3.29)(3.30)(3.31)(3.32)(3.33)(3.34)(3.35)(3.36)(3.37)(3.38)(3.39)(3.40) By hypothesis (i), (3.31) gives 𝛽=𝛼𝑦¹+𝑝. (3.41) Since ∇_𝑦2𝑦2𝑔(𝑥²,𝑦²) is positive or negative definite, (3.32) yields 𝛾=𝛿𝑦²+𝛼𝑟. (3.42) Suppose that 𝛼=0, then (3.42) implies 𝛾=𝛿𝑦². (3.43) Using (3.42) in (3.30), we get (𝛼−𝛿)∇_𝑦2𝑔𝑥²,𝑦²−𝑧²+∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝑟+12∇_𝑦2∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝑟𝛾−𝛿𝑦²=0, (3.44) which on using hypothesis (iii) and 𝛾=𝛿𝑦² yields 𝛼=𝛿. (3.45) As 𝛼=0, therefore the equations 𝛼=𝛿 and 𝛾=𝛿𝑦² give 𝛿=0 and 𝛾=0, respectively. Further, (3.41) implies 𝛽=0. Consequently, (𝛼,𝛽,𝛾,𝛿)=0, contradicting (3.40). Hence, we have 𝛼>0. (3.46) Subtracting (3.35) from (3.34) yields 𝛾−𝛿𝑦^2𝑇∇_𝑦2𝑔𝑥²,𝑦²−𝑧²+∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝑟=0. (3.47) Using (3.42) and (3.46) in the above equation, we get 𝑟^𝑇∇_𝑦2𝑔𝑥²,𝑦²−𝑧²+𝑟^𝑇∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝑟=0, (3.48) which contradicts hypothesis (ii) unless 𝑟=0. (3.49) Equation (3.42) yields 𝛾=𝛿𝑦². (3.50) Using (3.49) and (3.50) in (3.30), we obtain (𝛼−𝛿)∇_𝑦2𝑔𝑥²,𝑦²−𝑧²=0, (3.51) which on using hypothesis (iii) and (3.49) gives 𝛼=𝛿. (3.52) Since 𝛼>0, therefore 𝛿>0. (3.53) Now, using (3.41) and (3.46) in (3.29), we get ∇_𝑦1∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝𝑝=0, (3.54) which by hypothesis (iv) implies 𝑝=0. (3.55) By (3.41) and (3.55), we have 𝛽=𝛼𝑦¹. (3.56) Using (3.46), (3.55), and (3.56) in (3.27), we get ∇_𝑥1𝑓𝑥¹,𝑦¹+𝜂₁=0. (3.57) Equations (3.28), (3.46), (3.49), and (3.50) give ∇_𝑥2𝑔𝑥²,𝑦²+𝜂₂=0, (3.58) and hence, we also have 𝑥^2𝑇∇_𝑥2𝑔𝑥²,𝑦²+𝜂₂=0. (3.59) Thus, (𝑥¹,𝑦¹,𝑥²,𝑦²,𝑤¹=𝜂₁, 𝑤²=𝜂₂, 𝑞=0, 𝑠=0) satisfies the dual constraints from (3.7) to (3.10), and so it is a feasible solution for the dual problem (SMND).
Further, using (3.46), (3.55), and (3.56) in (3.33), we obtain 𝑦^1𝑇∇_𝑦1𝑓𝑥¹,𝑦¹=𝑦^1𝑇𝑧¹. (3.60) Moreover, since 𝛽=𝛼𝑦¹ and 𝛼>0, (3.36) implies 𝑦¹∈𝑁_𝐷1(𝑧¹) so that 𝑦^1𝑇𝑧¹=𝑆𝑦¹∣𝐷₁. (3.61) From (3.37), (3.50), (3.52), and (3.53), we get 𝑦²∈𝑁_𝐷2𝑧². (3.62) Since 𝐷₂ is a compact convex set in 𝑅^|𝐾2|, 𝑦^2𝑇𝑧²=𝑆𝑦²∣𝐷₂. (3.63) Therefore, using (3.38), (3.39), (3.49), (3.55), (3.57), and (3.60)–(3.63), we obtain 𝑓𝑥¹,𝑦¹+𝑆𝑥¹∣𝐶₁+𝑔𝑥²,𝑦²+𝑆𝑥²∣𝐶₂−𝑦^2𝑇𝑧²−𝑦^1𝑇∇_𝑦1𝑓𝑥¹,𝑦¹+∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝−12𝑝^𝑇∇_𝑦1𝑦1𝑓𝑥¹,𝑦¹𝑝−12𝑟^𝑇∇_𝑦2𝑦2𝑔𝑥²,𝑦²𝑟=𝑓𝑥¹,𝑦¹−𝑆𝑦¹∣𝐷₁+𝑔𝑥²,𝑦²−𝑆𝑦²∣𝐷₂+𝑥^2𝑇𝑤²−𝑥^1𝑇∇_𝑥1𝑓𝑥¹,𝑦¹+∇_𝑥1𝑥1𝑓𝑥¹,𝑦¹𝑞−12𝑞^𝑇∇_𝑥1𝑥1𝑓𝑥¹,𝑦¹𝑞−12𝑠^𝑇∇_𝑥2𝑥2𝑔𝑥²,𝑦²𝑠, (3.64) that is, the two objective function values are equal.
Finally, from Theorem 3.1, we get that (𝑥¹,𝑦¹,𝑥²,𝑦²,𝑧¹,𝑧²,𝑝,𝑟) and (𝑥¹,𝑦¹,𝑥²,𝑦²,𝑤¹,𝑤²,𝑞,𝑠) are global optimal solutions for (SMNP) and (SMND), respectively.

Theorem 3.3 (Converse duality). Let 𝑓∶𝑅^|𝐽1|×𝑅^|𝐾1|→𝑅 and 𝑔∶𝑅^|𝐽2|×𝑅^|𝐾2|→𝑅 be differentiable functions, and let (𝑢¹,𝑣¹,𝑢²,𝑣²,𝑤¹,𝑤²,𝑞,𝑠) be a local optimal solution of (SMND). Suppose that (i)the matrix ∇_𝑥1𝑥1𝑓(𝑢¹,𝑣¹) is non singular, (ii)∇_𝑥2𝑥2𝑔(𝑢²,𝑣²) is positive definite and 𝑠^𝑇(∇_𝑥2𝑔(𝑢²,𝑣²)+𝑤²)≥0 or ∇_𝑥2𝑥2𝑔(𝑢²,𝑣²) is negative definite and 𝑠^𝑇(∇_𝑥2𝑔(𝑢²,𝑣²)+𝑤²)≤0, (iii)∇_𝑥2𝑔(𝑢²,𝑣²)+𝑤²+∇_𝑥2𝑥2𝑔(𝑢²,𝑣²)𝑠≠0, (iv)one of the matrices (𝜕/𝜕𝑥¹_𝑖)(∇_𝑥1𝑥1𝑓(𝑢¹,𝑣¹)), 𝑖=1,2,…,|𝐽₁|, is positive or negative definite. Then, there exist 𝑧¹∈𝐷₁ and 𝑧²∈𝐷₂ such that (𝑢¹,𝑣¹,𝑢²,𝑣²,𝑧¹,𝑧²,𝑝=0,𝑟=0) is feasible for (SMNP) and the objective function values of (SMNP), and (SMND) are equal. Furthermore, if the assumptions of weak duality (Theorem 3.1) are satisfied for all feasible solutions of (SMNP) and (SMND), then (𝑢¹,𝑣¹,𝑢²,𝑣²,𝑤¹,𝑤²,𝑞,𝑠) and (𝑢¹,𝑣¹,𝑢²,𝑣²,𝑧¹,𝑧²,𝑝,𝑟) are global optimal solutions for (SMND) and (SMNP), respectively.

Proof. It follows on the lines of Theorem 3.2.

4. Special Cases

In this section, we consider some of the special cases of the problems studied in Section 3. (i)If 𝐽₂=∅ and 𝐾₂=∅, then our problems (SMNP) and (SMND) reduce to the programs (PP) and (DP) studied in Gulati and Gupta [17]. (ii)If 𝐽₂=∅, 𝐾₂=∅, 𝐶₁={0} and 𝐷₁={0}, then (SMNP) and (SMND) are reduced to the programs (SP) and (SD) studied in Gulati et al. [16] with the omission of nonnegativity constraints from (SP) and (SD). (iii)If 𝐽₂=∅, 𝐾₂=∅, 𝑝=0, and 𝑞=0, then (SMNP) and (SMND) become a pair of symmetric nondifferentiable dual programs considered in Mond and Schechter [6] with the omission of nonnegativity constraints from the programs (P) and (D) studied in Mond and Schechter. (iv)If 𝐽₂=∅, 𝐾₂=∅, 𝑝=0, 𝑞=0, 𝐶₁={0}, and 𝐷₁={0}, then the programs (WP) and (WD) of [15] are obtained with the omission of nonnegativity constraints from (WP) and (WD). (v)If 𝐽₁=∅ and 𝐾₁=∅ in (SMNP) and (SMND), then the programs studied in [10] are obtained. (vi)If 𝐽₁=∅, 𝐾₁=∅, 𝐶₂={0}, and 𝐷₂={0} in (SMNP) and (SMND), then the programs (SP1) and (SD1) of [16] are obtained with the omission of nonnegativity constraints from (SP1) and (SD1). (vii)If 𝐽₁=∅, 𝐾₁=∅, 𝑟=0, and 𝑠=0, then (SMNP) and (SMND) become a pair of symmetric nondifferentiable dual programs considered in [6] with the omission of nonnegativity constraints from the programs (P1) and (D1) studied in Mond and Schechter.(viii)If 𝐽₁=∅, 𝐾₁=∅, 𝑟=0, 𝑠=0, 𝐶₂={0}, and 𝐷₂={0}, then (SMNP) and (SMND) become a pair of single objective symmetric differentiable dual programs considered in [15] with the omission of nonnegativity constraints from (MP) and (MD). (ix)By eliminating the second-order and nondifferentiable terms, our problems (SMNP) and (SMND) reduce to the mixed symmetric dual programs studied by Chandra et al. [11] with the omission of 𝑥¹>0, 𝑥²>0, 𝑣¹>0, and 𝑣²>0 from the programs studied in Chandra et al. [11]. (x)By eliminating the second-order terms, our problems are reduced to the programs (MP) and (MD) studied in [12] with the omission of nonnegativity constraints from (MP) and (MD).

5. Concluding Remarks

It is to be noted that previously known results [6, 10–12, 15–17] are special cases of our study. It is not clear whether the second-order mixed symmetric duality in mathematical programming can be further extended to higher-order multiobjective symmetric dual programs formulated in [19].

Acknowledgments

Ravi P. Agarwal and Izhar Ahmad are thankful to the King Fahd University of Petroleum and Minerals, Saudi Arabia for providing financial support to carry out this research.