On Second-Order Duality for Minimax Fractional Programming Problems with Generalized Convexity

Abstract

We focus our study on a discussion of duality relationships of a minimax fractional programming problem with its two types of second-order dual models under the second-order generalized convexity type assumptions. Results obtained in this paper naturally unify and extend some previously known results on minimax fractional programming in the literature.

1. Introduction

Fractional programming is an interesting subject applicable to many types of optimization problems such as portfolio selection, production, and information theory and numerous decision making problems in management science. More specifically, it can be used in engineering and economics to minimize a ratio of physical or economical functions, or both, such as cost/time, cost/volume, and cost/benefit, in order to measure the efficiency or productivity of the system (see Stancu-Minasian [1]).

Minimax type functions arise in the design of electronic circuits; however, minimax fractional problems appear in the formulation of discrete and continuous rational approximation problems with respect to the Chebyshev norm [2], continuous rational games [3], multiobjective programming [4, 5], and engineering design as well as some portfolio selection problems discussed by Bajona-Xandri and Martinez-Legaz [6].

In this paper, we consider the minimax fractional programming problemminimize𝜙(𝑥)=sup_𝑦∈𝑌𝑓(𝑥,𝑦)subjectto𝑔(𝑥)≤0,ℎ𝑥(∈𝑥,𝑅𝑦)_𝑛,,(1.1) where 𝑌 is a compact subset of 𝑅^𝑙 and 𝑓(⋅,⋅)∶𝑅^𝑛×𝑅^𝑙→𝑅,  ℎ(⋅,⋅)∶𝑅^𝑛×𝑅^𝑙→𝑅, and 𝑔(⋅)∶𝑅^𝑛→𝑅^𝑚 are twice continuously differentiable functions on 𝑅^𝑛×𝑅^𝑙, 𝑅^𝑛×𝑅^𝑙, and 𝑅^𝑛, respectively. It is assumed that, for each (𝑥,𝑦) in 𝑅^𝑛×𝑅^𝑙, 𝑓(𝑥,𝑦)≥0 and ℎ(𝑥,𝑦)>0.

For the case of convex differentiable minimax fractional programming, Yadav and Mukherjee [7] formulated two dual models for (1.1) and derived duality theorems. Chandra and Kumar [8] pointed out certain omissions and inconsistencies in the dual formulation of Yadav and Mukherjee [7]; they constructed two modified dual problems for (1.1) and proved appropriate duality results. Liu and Wu [9, 10] and Ahmad [11] obtained sufficient optimality conditions and duality theorems for (1.1) assuming the functions involved to be generalized convex.

Second-order duality provides tighter bounds for the value of the objective function when approximations are used. For more details, one can consult ([12, page 93]). One more advantage of second-order duality, when applicable, is that, if a feasible point in the primal is given and first-order duality does not apply, then we can use second order duality to provide a lower bound of the value of the primal (see [13]).

Mangasarian [14] first formulated the second-order dual for a nonlinear programming problem and established second-order duality results under certain inequalities. Mond [12] reproved second-order duality results assuming rather simple inequalities. Subsequently, Bector and Chandra [15] formulated a second-order dual for a fractional programming problem and obtained usual duality results under the assumptions [14] by naming these as convex/concave functions.

Based upon the ideas of Bector et al. [16] and Rueda et al. [17], Yang and Hou [18] proposed a new concept of generalized convexity and discussed sufficient optimality conditions for (1.1) and duality results for its corresponding dual. Recently, Husain et al. [19] formulated two types of second-order dual models to (1.1) and discussed appropriate duality results involving 𝜂-convexity/generalized 𝜂-convexity assumptions.

In this paper, we are inspired by Chandra and Kumar [8], Bector et al. [16], Liu [20], and Husain et al. [19] to discuss weak, strong, and strict converse duality theorems connecting (1.1) with its two types of second-order duals by using second-order generalized convexity type assumptions [21].

2. Notations and Preliminaries

Let 𝑆={𝑥∈𝑅^𝑛∶𝑔(𝑥)≤0} denote the set of all feasible solutions of (1.1). For each (𝑥,𝑦)∈𝑅^𝑛×𝑅^𝑙, we define 𝐽(𝑥)=𝑗∈𝑀∶𝑔_𝑗(𝑥)=0,(2.1) where 𝑀={1,2,…,𝑚},𝐾𝑌((𝑥𝑥))==𝑦(𝑠∈,𝑡𝑌,𝑦∶)̃𝑓∈(ℕ𝑥,×𝑦)𝑅=_𝑠+sup_𝑧∈𝑌𝑓(𝑥,𝑧),×𝑅^𝑙𝑠∶1≤𝑠≤𝑛+1,𝑡=𝑡₁,𝑡₂,…,𝑡_𝑠∈𝑅^𝑠+with^𝑠_𝑖=1𝑡_𝑖=1,𝑦̃=𝑦₁,𝑦₂,….𝑦_𝑠with𝑦_𝑖∈𝑌(𝑥),𝑖=1,2,…,𝑠.(2.2)

Definition 2.1. A functional ℱ∶𝑋×𝑋×𝑅^𝑛→𝑅, where 𝑋⊆𝑅^𝑛 is said to be sublinear in its third argument, if ∀𝑥,𝑥∈𝑋,(i)ℱ(𝑥,𝑥;𝑎₁+𝑎₂)≤ℱ(𝑥,𝑥;𝑎₁)+ℱ(𝑥,𝑥;𝑎₂)∀𝑎₁,𝑎₂∈𝑅^𝑛, (ii)ℱ(𝑥,𝑥;𝛼𝑎)=𝛼ℱ(𝑥,𝑥;𝑎)∀𝛼∈𝑅₊,𝑎∈𝑅^𝑛. By (ii), it is clear that ℱ(𝑥,𝑥;0𝑎)=0.

Definition 2.2. A point 𝑥∈𝑆 is said to optimal solution of (1.1) if 𝜙(𝑥)≥𝜙(𝑥) for each 𝑥∈𝑆.

The following theorem [8] will be needed in the subsequent analysis.

Theorem 2.3 (necessary conditions). Let 𝑥^∗ be a solution (local or global) of (1.1), and let ∇𝑔_𝑗(𝑥^∗),𝑗∈𝐽(𝑥^∗) be linearly independent. Then there exist (𝑠^∗,𝑡^∗,𝑦^∗)∈𝐾(𝑥^∗),𝜆^∗∈𝑅₊, and 𝜇^∗∈𝑅^𝑚+ such that ∇^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑥^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑥^∗,𝑦^∗𝑖+∇^𝑚_𝑗=1𝑓𝑥𝜇^∗𝑗𝑔_𝑗𝑥^∗=0,_∗,𝑦^∗𝑖−𝜆^∗ℎ𝑥^∗,𝑦^∗𝑖=0,𝑖=1,2,…,𝑠^∗,_𝑚_𝑗=1𝑡𝜇^∗𝑗𝑔_𝑗𝑥^∗=0,_∗𝑖≥0,^𝑠∗_𝑖=1𝑡^∗𝑖=1,𝑦^∗𝑖∈𝑌𝑥^∗,𝑖=1,2,…,𝑠^∗.(2.3) Throughout the paper, we assume that ℱ is a sublinear functional. For 𝛽=1,2,…,𝑟 let 𝑏,𝑏₀,𝑏_𝛽∶𝑋×𝑋→𝑅₊,𝜙,𝜙₀,𝜙_𝛽∶𝑅→𝑅,𝜌,𝜌₀,𝜌_𝛽 be real numbers, and let 𝜃∶𝑅^𝑛×𝑅^𝑛→𝑅.

3. First Duality Model

In this section, we discuss usual duality results for the following dual [19]:max_{𝑠,𝑡,𝑦∈𝐾(𝑧)}sup_{(𝑧,𝜇,𝜆,𝑝)∈𝐻1𝑠,𝑡,𝑦}𝜆,(3.1) where 𝐻₁(𝑠,𝑡,𝑦) denotes the set of all (𝑧,𝜇,𝜆,𝑝)∈𝑅^𝑛×𝑅^𝑚+×𝑅₊×𝑅^𝑛 satisfying∇^𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖+∇^2𝑠_𝑖=1+∇_𝑚𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖𝑝_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)+∇^2𝑚_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)𝑝=0,(3.2)_𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖−12𝑝^𝑇∇^2𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖𝑝≥0,(3.3)_𝑚_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)−12𝑝^𝑇∇^2𝑚_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)𝑝≥0.(3.4) If, for a triplet (𝑠,𝑡,𝑦)∈𝐾(𝑧), the set 𝐻₁(𝑠,𝑡,𝑦)=∅, then we define the supremum over it to be −∞.

Remark 3.1. If 𝑃=0, then (3.1) becomes the dual considered in [9].

Theorem 3.2 (weak duality). Let 𝑥 and (𝑧,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝) be the feasible solutions of (1.1) and (3.1), respectively. Suppose that there exist ℱ,𝜃,𝜙,𝑏 and 𝜌 such that 𝑏(𝑥,𝑧)𝜙^𝑠_𝑖=1𝑡_𝑖𝑓𝑥,𝑦_𝑖−𝜆ℎ𝑥,𝑦_𝑖−^𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖−^𝑚_𝑗=1+1𝜇_𝑗𝑔_𝑗(𝑧)2𝑝^𝑇∇^2𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖𝑝+12𝑝^𝑇𝑚_𝑗=1⟹ℱ𝑥,𝑧;∇_𝑠𝜇_𝑗𝑔_𝑗(𝑧)𝑝<0_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖+∇^𝑚_𝑗=1+∇_2𝑠𝜇_𝑗𝑔_𝑗(𝑧)_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖𝑝+∇^2𝑚_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)𝑝<−𝜌‖𝜃(𝑥,𝑧)‖².(3.5) Further assume that 𝑎<𝑏0(⟹𝑥𝜌,𝑧≥)𝜙0>.(𝑎0),<0,(((333...678))) Then sup_𝑦∈𝑌𝑓(𝑥,𝑦)ℎ(𝑥,𝑦)≥𝜆.(3.9)

Proof. Suppose contrary to the result that sup_𝑦∈𝑌𝑓(𝑥,𝑦)ℎ(𝑥,𝑦)<𝜆.(3.10) Thus, we have 𝑓𝑥,𝑦_𝑖−𝜆ℎ𝑥,𝑦_𝑖<0,∀𝑦_𝑖∈𝑌(𝑥),𝑖=1,2,…,𝑠.(3.11) It follows from 𝑡_𝑖≥0,𝑖=1,2,…,𝑠, that 𝑡_𝑖𝑓𝑥,𝑦_𝑖−𝜆ℎ𝑥,𝑦_𝑖≤0,(3.12) with at least one strict inequality since 𝑡=(𝑡₁,𝑡₂,…,𝑡_𝑠)≠0. Taking summation over 𝑖, we have ^𝑠_𝑖=1𝑡_𝑖𝑓𝑥,𝑦_𝑖−𝜆ℎ𝑥,𝑦_𝑖<0,(3.13) which together with (3.3) gives ^𝑠_𝑖=1𝑡_𝑖𝑓𝑥,𝑦_𝑖−𝜆ℎ𝑥,𝑦_𝑖≤<0_𝑠_𝑖=1−1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖2𝑝^𝑇∇^2𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖𝑝.(3.14) The above inequality along with (3.4) implies ^𝑠_𝑖=1𝑡_𝑖𝑓𝑥,𝑦_𝑖−𝜆ℎ𝑥,𝑦_𝑖−^𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖−^𝑚_𝑗=1+1𝜇_𝑗𝑔_𝑗(𝑧)2𝑝^𝑇∇^2𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖𝑝+12𝑝^𝑇∇^2𝑚_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)𝑝<0.(3.15) Using (3.6) and (3.7), it follows from (3.15) that 𝑏(𝑥,𝑧)𝜙^𝑠_𝑖=1𝑡_𝑖𝑓𝑥,𝑦_𝑖−𝜆ℎ𝑥,𝑦_𝑖−^𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖−^𝑚_𝑗=1+1𝜇_𝑗𝑔_𝑗(𝑧)2𝑝^𝑇∇^2𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖𝑝+12𝑝^𝑇∇^2𝑚_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)𝑝<0,(3.16) which along with (3.5) and (3.8) yields ℱ𝑥,𝑧;∇^𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖+∇^𝑚_𝑗=1+∇_2𝑠𝜇_𝑗𝑔_𝑗(𝑧)_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖𝑝+∇^2𝑚_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)𝑝<0,(3.17) which contradicts (3.2) since ℱ(𝑥,𝑧;0)=0.

Theorem 3.3 (strong duality). Assume that 𝑥^∗ is an optimal solution of (1.1) and ∇𝑔_𝑗(𝑥^∗),𝑗∈𝐽(𝑥^∗) are linearly independent. Then there exist (𝑠^∗,𝑡^∗,𝑦^∗)∈𝐾(𝑥^∗) and (𝑥^∗,𝜇^∗,𝜆^∗,𝑝^∗=0)∈𝐻₁(𝑠^∗,𝑡^∗,𝑦^∗) such that (𝑥^∗,𝜇^∗,𝜆^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑝^∗=0) is a feasible solution of (3.1) and the two objectives have the same values. Further, if the assumptions of weak duality (Theorem 3.2) hold for all feasible solutions (𝑧,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝) of (3.1), then (𝑥^∗,𝜇^∗,𝜆^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑝^∗=0) is an optimal solution of (3.1).

Proof. Since 𝑥^∗ is an optimal solution of (1.1) and ∇𝑔_𝑗(𝑥^∗),𝑗∈𝐽(𝑥^∗) are linearly independent, then, by Theorem 2.3, there exist (𝑠^∗,𝑡^∗,𝑦^∗)∈𝐾(𝑥^∗) and (𝑥^∗,𝜇^∗,𝜆^∗,𝑝^∗=0)∈𝐻₁(𝑠^∗,𝑡^∗,𝑦^∗) such that (𝑥^∗,𝜇^∗,𝜆^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑝^∗=0) is a feasible solution of (3.1) and the two objectives have the same values. Optimality of (𝑥^∗,𝜇^∗,𝜆^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑝^∗=0) for (3.1) thus follows from weak duality (Theorem 3.2).

Theorem 3.4 (Strict converse duality). Let 𝑥^∗ and (𝑧^∗,𝜇^∗,𝜆^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑝^∗) be the optimal solutions of (1.1) and (3.1), respectively. Suppose that ∇𝑔_𝑗(𝑥^∗),𝑗∈𝐽(𝑥^∗) are linearly independent and there exist ℱ,𝜃,𝜙,𝑏 and 𝜌 such that 𝑏𝑥^∗,𝑧^∗𝜙⎡⎢⎢⎣^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑥^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑥^∗,𝑦^∗𝑖−^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖−^𝑚_𝑗=1+1𝜇^∗𝑗𝑔_𝑗𝑧^∗2𝑝^∗𝑇∇^2𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖𝑝^∗+12𝑝^∗𝑇∇^2𝑚_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑧^∗𝑝^∗⟹ℱ⎛⎜⎜⎝𝑥_∗⎤⎥⎥⎦≤0,𝑧^∗;∇^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖+∇^𝑚_𝑗=1+∇_2𝑠∗𝜇^∗𝑗𝑔_𝑗𝑧^∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖𝑝^∗+∇^2𝑚_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑧^∗𝑝^∗⎞⎟⎟⎠<−𝜌‖‖𝜃𝑥_∗,𝑧^∗‖‖².(3.18) Further Assume 𝑎<𝑏0𝑥⟹𝜙(𝑎)≤0,(3.19)_∗𝜌,𝑧≥^∗0.>0,((33..2201)) Then 𝑧^∗=𝑥^∗, that is, 𝑧^∗ is an optimal solution of (1.1).

Proof. Suppose contrary to the result that 𝑧^∗≠𝑥^∗. Since 𝑥^∗ and (𝑧^∗,𝜇^∗,𝜆^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑝^∗) are optimal solutions of (1.1) and (3.1), respectively, and ∇𝑔_𝑗(𝑥^∗),𝑗∈𝐽(𝑥^∗) are linearly independent, therefore, from strong duality (Theorem 3.3), we reach sup_{𝑦∗∈𝑌}𝑓𝑥^∗,𝑦^∗ℎ(𝑥^∗,𝑦^∗)=𝜆^∗.(3.22) Thus, we have 𝑓𝑥^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑥^∗,𝑦^∗𝑖≤0,∀𝑦^∗𝑖∈𝑌𝑥^∗,𝑖=1,2,…,𝑠^∗.(3.23) Now, proceeding as in Theorem 3.2, we get ^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑥^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑥^∗,𝑦^∗𝑖−^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖−^𝑚_𝑗=1+1𝜇^∗𝑗𝑔_𝑗𝑧^∗2𝑝^∗𝑇∇^2𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖𝑝^∗+12𝑝^∗𝑇∇^2𝑚_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑧^∗𝑝^∗<0.(3.24) Using (3.19) and (3.20), it follows from (3.24) that 𝑏𝑥^∗,𝑧^∗𝜙⎡⎢⎢⎣^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑥^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑥^∗,𝑦^∗𝑖−^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖−^𝑚_𝑗=1+1𝜇^∗𝑗𝑔_𝑗𝑧^∗2𝑝^∗𝑇∇^2𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖𝑝^∗+12𝑝^∗𝑇∇^2𝑚_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑧^∗𝑝^∗⎤⎥⎥⎦≤0,(3.25) which along with (3.18) and (3.21) implies ℱ⎛⎜⎜⎝𝑥^∗,𝑧^∗;∇^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖+∇^𝑚_𝑗=1+∇_2𝑠∗𝜇^∗𝑗𝑔_𝑗𝑧^∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖𝑝^∗+∇^2𝑚_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑧^∗𝑝^∗⎞⎟⎟⎠<0,(3.26) which contradicts (3.2) since ℱ(𝑥^∗,𝑧^∗;0)=0.

4. Second Duality Model

This section deals with duality theorems for the following second-order dual to (1.1):max_{(𝑠,𝑡,𝑦)∈𝐾(𝑧)}sup_{(𝑧,𝜇,𝜆,𝑝)∈𝐻2(𝑠,𝑡,𝑦)}𝜆,(4.1) where 𝐻₂(𝑠,𝑡,𝑦) denotes the set of all (𝑧,𝜇,𝜆,𝑝)∈𝑅^𝑛×𝑅^𝑚+×𝑅₊×𝑅^𝑛 satisfying∇∑^𝑠_𝑖=1+𝑡_𝑖∇𝑓_𝑚𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖+∇∑^2𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖𝑝_𝑗∑₌₁𝜇_𝑗𝑔_𝑗(𝑧)+∇∑^2𝑚_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)𝑝=0,(4.2)_𝑠𝑖∑₌₁𝑡_𝑖−𝑓1𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖+∑_{𝑗∈𝐽∘}𝜇_𝑗𝑔_𝑗(𝑧)2𝑝^𝑇∇²^𝑠∑_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖+∑_{𝑗∈𝐽∘}𝜇_𝑗𝑔_𝑗(𝑧)𝑝≥0,(4.3)_{𝑗∈𝐽𝛼}𝜇_𝑗𝑔_𝑗(𝑧)−12𝑝^𝑇∇²_𝑗_∈𝐽𝛼𝜇_𝑗𝑔_𝑗(𝑧)𝑝≥0,𝛼=1,2,…,𝑟,(4.4) where 𝐽_𝛼⊆𝑀,𝛼=0,1,2,…,𝑟, with ⋃^𝑟𝛼=0𝐽_𝛼=𝑀 and 𝐽_𝛼∩𝐽_𝛽=∅, if 𝛼≠𝛽.

If, for a triplet (𝑠,𝑡,𝑦)∈𝐾(𝑧), the set 𝐻₂(𝑠,𝑡,𝑦)=∅, then we define the supremum over it to be −∞.

Theorem 4.1 (weak duality). Let 𝑥 and (𝑧,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝) be the feasible solutions of (1.1) and (4.1), respectively. Suppose that there exist ℱ,𝜃,𝜙₀,𝑏₀,𝜌₀ and 𝜙_𝛽,𝑏_𝛽,𝜌_𝛽,𝛽=1,2,…,𝑟 such that 𝑏₀(𝑥,𝑧)𝜙₀⎡⎢⎢⎣^𝑠_𝑖=1𝑡_𝑖𝑓𝑥,𝑦_𝑖−𝜆ℎ𝑥,𝑦_𝑖−^𝑠_𝑖=1+1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖−_𝑗∈𝐽0𝜇_𝑗𝑔_𝑗(𝑧)2𝑝^𝑇∇²⎛⎜⎜⎝^𝑠_𝑖=1⟹ℱ𝑥,𝑧;∇_𝑠𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖+_𝑗∈𝐽0𝜇_𝑗𝑔_𝑗(𝑧)⎞⎟⎟⎠𝑝⎤⎥⎥⎦<0_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖+∇^2𝑠_𝑖=1+∇_𝑗∈𝐽0𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖𝑝𝜇_𝑗𝑔_𝑗(𝑧)+∇²_𝑗_∈𝐽0−𝑏_𝛼𝜇_𝑗𝑔_𝑗(𝑧)𝑝⎞⎟⎟⎠<−𝜌₀‖𝜃(𝑥,𝑧)‖²,(4.5)(𝑥,𝑧)𝜙_𝛼⎡⎢⎢⎣_{𝑗∈𝐽𝛼}𝜇_𝑗𝑔_𝑗(𝑧)−12𝑝^𝑇∇²_𝑗_∈𝐽𝛼⟹ℱ⎛⎜⎜⎝𝑥,𝑧;∇_{𝑗∈𝐽𝛼}𝜇_𝑗𝑔_𝑗(𝑧)𝑝⎤⎥⎥⎦≤0𝜇_𝑗𝑔_𝑗(𝑧)+∇²_𝑗_∈𝐽𝛼𝜇_𝑗𝑔_𝑗(𝑧)𝑝⎞⎟⎟⎠≤−𝜌_𝛼‖𝜃(𝑥,𝑧)‖²,𝛼=1,2,…,𝑟.(4.6) Further assume that 𝑎≥0⟹𝑎<𝜙_𝛼0(⟹𝑎)≥𝜙0₀,𝛼=1,2,…,𝑟,(4.7)𝑏(𝑎)<0,(4.8)₀(𝑥,𝑧)>0,𝜌𝑏_𝛼(𝑥,𝑧)≥0,𝛼=1,2,…,𝑟,(4.9)₀+^𝑟_𝛼=1𝜌_𝛼≥0.s(4.10) Then sup_𝑦∈𝑌𝑓(𝑥,𝑦)ℎ(𝑥,𝑦)≥𝜆.(4.11)

Proof. Suppose contrary to the result that sup_𝑦∈𝑌𝑓(𝑥,𝑦)ℎ(𝑥,𝑦)<𝜆.(4.12) Thus, we have 𝑓𝑥,𝑦_𝑖−𝜆ℎ𝑥,𝑦_𝑖<0,∀𝑦_𝑖∈𝑌(𝑥),𝑖=1,2,…,𝑠.(4.13) It follows from 𝑡_𝑖≥0,𝑖=1,2,…,𝑠, that 𝑡_𝑖𝑓𝑥,𝑦_𝑖−𝜆ℎ𝑥,𝑦_𝑖≤0,(4.14) with at least one strict inequality since 𝑡=(𝑡₁,𝑡₂,…,𝑡_𝑠)≠0. Taking summation over 𝑖, we have ^𝑠_𝑖=1𝑡_𝑖𝑓𝑥,𝑦_𝑖−𝜆ℎ𝑥,𝑦_𝑖<0,(4.15) which together with (4.3) implies ^𝑠_𝑖=1𝑡_𝑖𝑓𝑥,𝑦_𝑖−𝜆ℎ𝑥,𝑦_𝑖≤<0_𝑠_𝑖=1−1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖+_𝑗∈𝐽0𝜇_𝑗𝑔_𝑗(𝑧)2𝑝^𝑇∇²⎡⎢⎢⎣^𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖+_𝑗∈𝐽0𝜇_𝑗𝑔_𝑗(𝑧)⎤⎥⎥⎦𝑝.(4.16) Using (4.8) and (4.9), it follows from (4.16) that 𝑏₀(𝑥,𝑧)𝜙₀⎡⎢⎢⎣^𝑠_𝑖=1𝑡_𝑖𝑓𝑥,𝑦_𝑖−𝜆ℎ𝑥,𝑦_𝑖−^𝑠_𝑖=1+1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖−_𝑗∈𝐽0𝜇_𝑗𝑔_𝑗(𝑧)2𝑝^𝑇∇²⎡⎢⎢⎣^𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖+_𝑗∈𝐽0𝜇_𝑗𝑔_𝑗(𝑧)⎤⎥⎥⎦𝑝⎤⎥⎥⎦<0,(4.17) which by (4.5) implies ℱ𝑥,𝑧;∇^𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖+∇^2𝑠_𝑖=1+∇_𝑗∈𝐽0𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖𝑝𝜇_𝑗𝑔_𝑗(𝑧)+∇²_𝑗_∈𝐽0𝜇_𝑗𝑔_𝑗(𝑧)𝑝⎞⎟⎟⎠<−𝜌₀‖𝜃(𝑥,𝑧)‖².(4.18) Also, inequality (4.4) along with (4.7) and (4.9) yields −𝑏_𝛼(𝑥,𝑧)𝜙_𝛼⎡⎢⎢⎣_{𝑗∈𝐽𝛼}𝜇_𝑗𝑔_𝑗(𝑧)−12𝑝^𝑇∇²_𝑗_∈𝐽𝛼𝜇_𝑗𝑔_𝑗(𝑧)𝑝⎤⎥⎥⎦≤0,𝛼=1,2,…,𝑟.(4.19) From (4.6) and the above inequality, we have ℱ⎛⎜⎜⎝𝑥,𝑧;∇_{𝑗∈𝐽𝛼}𝜇_𝑗𝑔_𝑗(𝑧)+∇²_𝑗_∈𝐽𝛼𝜇_𝑗𝑔_𝑗(𝑧)𝑝⎞⎟⎟⎠≤−𝜌_𝛼‖𝜃(𝑥,𝑧)‖²,𝛼=1,2,…,𝑟.(4.20) On adding (4.18) and (4.20) and making use of the sublinearity of ℱ with (4.10), we obtain ℱ𝑥,𝑧;∇^𝑠_𝑖=1𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖+∇^2𝑠_𝑖=1+∇_𝑚𝑡_𝑖𝑓𝑧,𝑦_𝑖−𝜆ℎ𝑧,𝑦_𝑖𝑝_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)+∇^2𝑚_𝑗=1𝜇_𝑗𝑔_𝑗(𝑧)𝑝<0,(4.21) which contradicts (4.2) since ℱ(𝑥,𝑧;0)=0.

The proof of the following theorem is similar to that of Theorem 3.3 and, hence, is omitted.

Theorem 4.2 (strong duality). Assume that 𝑥^∗ is an optimal solution of (1.1) and ∇𝑔_𝑗(𝑥^∗),𝑗∈𝐽(𝑥^∗), are linearly independent. Then there exist (𝑠^∗,𝑡^∗,𝑦^∗)∈𝐾(𝑥^∗) and (𝑥^∗,𝜇^∗,𝜆^∗,𝑝^∗=0)∈𝐻₂(𝑠^∗,𝑡^∗,𝑦^∗) such that (𝑥^∗,𝜇^∗,𝜆^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑝^∗=0) is a feasible solution of (4.1) and the two objectives have the same values. Further, if the assumptions of weak duality (Theorem 4.1) hold for all feasible solutions (𝑧,𝜇,𝜆,𝑠,𝑡,𝑦,𝑝) of (4.1), then (𝑥^∗,𝜇^∗,𝜆^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑝^∗=0) is an optimal solution of (4.1).

Theorem 4.3 (strict converse duality). Let 𝑥^∗ and (𝑧^∗,𝜇^∗,𝜆^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑝^∗) be the optimal solutions of (1.1) and (4.1), respectively. Suppose that ∇𝑔_𝑗(𝑥^∗),𝑗∈𝐽(𝑥^∗) are linearly independent and there exist ℱ,𝜃,𝜙₀,𝑏₀,𝜌₀ and 𝜙_𝛽,𝑏_𝛽,𝜌_𝛽,𝛽=1,2,…,𝑟 such that 𝑏₀𝑥^∗,𝑧^∗𝜙₀⎡⎢⎢⎣^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑥^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑥^∗,𝑦^∗𝑖−^𝑠∗_𝑖=1−𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖_𝑗∈𝐽0𝜇^∗𝑗𝑔_𝑗𝑧^∗+12𝑝^∗𝑇∇²⎛⎜⎜⎝^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖+_𝑗∈𝐽0𝜇^∗𝑗𝑔_𝑗𝑧^∗⎞⎟⎟⎠𝑝^∗⟹ℱ⎛⎜⎜⎝𝑥_∗⎤⎥⎥⎦≤0,𝑧^∗;∇^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖+∇^2𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖𝑝^∗+∇_𝑗∈𝐽0𝜇^∗𝑗𝑔_𝑗𝑧^∗+∇²_𝑗_∈𝐽0𝜇^∗𝑗𝑔_𝑗𝑧^∗𝑝^∗⎞⎟⎟⎠<−𝜌₀‖‖𝜃𝑥^∗,𝑧^∗‖‖²−𝑏_𝛼(4.22)𝑥^∗,𝑧^∗𝜙_𝛼⎡⎢⎢⎣_{𝑗∈𝐽𝛼}𝜇^∗𝑗𝑔_𝑗𝑧^∗−12𝑝^∗𝑇∇²_𝑗_∈𝐽𝛼𝜇^∗𝑗𝑔_𝑗𝑧^∗𝑝^∗⟹ℱ⎛⎜⎜⎝𝑥_∗⎤⎥⎥⎦≤0,𝑧^∗;∇_{𝑗∈𝐽𝛼}𝜇^∗𝑗𝑔_𝑗𝑧^∗+∇²_𝑗_∈𝐽𝛼𝜇^∗𝑗𝑔_𝑗𝑧^∗𝑝^∗⎞⎟⎟⎠≤−𝜌_𝛼‖‖𝜃𝑥^∗,𝑧^∗‖‖²,𝛼=1,2,…,𝑟.(4.23) Further assume that 𝑎≥0⟹𝑎<𝜙_𝛼0(⟹𝑎)≥𝜙0₀,𝛼=1,2,…,𝑟,(4.24)𝑏(𝑎)≤0,(4.25)₀𝑥^∗,𝑧^∗>0,𝜌𝑏_𝛼𝑥^∗,𝑧^∗≥0,𝛼=1,2,…,𝑟,(4.26)₀+^𝑟_𝛼=1𝜌_𝛼≥0.(4.27) Then 𝑧^∗=𝑥^∗, that is, 𝑧^∗ is an optimal solution of (1.1).

Proof. Suppose contrary to the result that 𝑧^∗≠𝑥^∗. Since 𝑥^∗ and (𝑧^∗,𝜇^∗,𝜆^∗,𝑠^∗,𝑡^∗,𝑦^∗,𝑝^∗) are optimal solutions of (1.1) and (4.1), respectively, and ∇𝑔_𝑗(𝑥^∗),𝑗∈𝐽(𝑥^∗) are linearly independent, therefore, from strong duality (Theorem 4.2), we reach sup_{𝑦∗∈𝑌}𝑓𝑥^∗,𝑦^∗ℎ(𝑥^∗,𝑦^∗)=𝜆^∗.(4.28) Thus, we have 𝑓𝑥^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑥^∗,𝑦^∗𝑖≤0,∀𝑦^∗𝑖∈𝑌𝑥^∗,𝑖=1,2,…,𝑠^∗.(4.29) Now, proceeding as in Theorem 4.1, we get ^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑥^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑥^∗,𝑦^∗𝑖−^𝑠∗_𝑖=1−𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖_𝑗∈𝐽0𝜇^∗𝑗𝑔_𝑗𝑧^∗+12𝑝^∗𝑇∇²⎛⎜⎜⎝^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖+_𝑗∈𝐽0𝜇^∗𝑗𝑔_𝑗𝑧^∗⎞⎟⎟⎠𝑝^∗<0.(4.30) Using (4.25) and (4.26), it follows from (4.30) that 𝑏₀𝑥^∗,𝑧^∗𝜙₀⎡⎢⎢⎣^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑥^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑥^∗,𝑦^∗𝑖−^𝑠∗_𝑖=1+1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖−_𝑗∈𝐽0𝜇^∗𝑗𝑔_𝑗𝑧^∗2𝑝^∗𝑇∇²⎛⎜⎜⎝^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖+_𝑗∈𝐽0𝜇^∗𝑗𝑔_𝑗𝑧^∗⎞⎟⎟⎠𝑝^∗⎤⎥⎥⎦≤0,(4.31) which by (4.22) implies ℱ⎛⎜⎜⎝𝑥^∗,𝑧^∗;∇^𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖+∇^2𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖𝑝^∗+∇_𝑗∈𝐽0𝜇^∗𝑗𝑔_𝑗𝑧^∗+∇²_𝑗_∈𝐽0𝜇^∗𝑗𝑔_𝑗𝑧^∗𝑝^∗⎞⎟⎟⎠<−𝜌₀‖‖𝜃𝑥^∗,𝑧^∗‖‖².(4.32) Also, inequality (4.4) along with (4.24) and (4.26) yields −𝑏_𝛼𝑥^∗,𝑧^∗𝜙_𝛼⎡⎢⎢⎣_{𝑗∈𝐽𝛼}𝜇^∗𝑗𝑔_𝑗𝑧^∗−12𝑝^∗𝑇∇²_𝑗_∈𝐽𝛼𝜇^∗𝑗𝑔_𝑗𝑧^∗𝑝^∗⎤⎥⎥⎦≤0,𝛼=1,2,…,𝑟.(4.33) From (4.23) and the above inequality, we have ℱ⎛⎜⎜⎝𝑥^∗,𝑧^∗;∇_{𝑗∈𝐽𝛼}𝜇^∗𝑗𝑔_𝑗𝑧^∗+∇²_𝑗_∈𝐽𝛼𝜇^∗𝑗𝑔_𝑗𝑧^∗𝑝^∗⎞⎟⎟⎠≤−𝜌_𝛼‖𝜃(𝑥^∗,𝑧^∗)‖²,𝛼=1,2,…,𝑟.(4.34) On adding (4.32) and (4.34) and making use of the sublinearity of ℱ with (4.27), we obtain ℱ⎛⎜⎜⎝𝑥^∗,𝑧^∗;∇^𝑠∗_𝑖=1𝑡^∗_i𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖+∇^2𝑠∗_𝑖=1𝑡^∗𝑖𝑓𝑧^∗,𝑦^∗𝑖−𝜆^∗ℎ𝑧^∗,𝑦^∗𝑖𝑝^∗+∇^𝑚_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑧^∗+∇^2𝑚_𝑗=1𝜇^∗𝑗𝑔_𝑗𝑧^∗𝑝^∗<0,(4.35) which contradicts (4.2) since ℱ(𝑥^∗,𝑧^∗;0)=0.

5. Conclusion and Further Developments

In this paper, we have established weak, strong, and strict converse duality theorems for a class of minimax fractional programming problems in the frame work of second-order generalized convexity. The second-order duality results developed in this paper can be further extended for the following nondifferentiable minimax fractional programming problem [22, 23]:minimize𝜓(𝑥)=sup_𝑦∈𝑌𝑓(𝑥,𝑦)+𝑥^𝑇𝐵𝑥^1/2ℎ(𝑥,𝑦)−𝑥^𝑇𝐷𝑥^1/2subjectto𝑔(𝑥)≤0,𝑥∈𝑅_𝑛,,(5.1) where 𝑌 is a compact subset of 𝑅^𝑙, 𝐵 and 𝐷 are 𝑛×𝑛 positive semidefinite symmetric matrices, and 𝑓(⋅,⋅)∶𝑅^𝑛×𝑅^𝑙→𝑅,   ℎ(⋅,⋅)∶𝑅^𝑛×𝑅^𝑙→𝑅, and 𝑔(⋅)∶𝑅^𝑛→𝑅^𝑚 are twice continuously differentiable functions on 𝑅^𝑛×𝑅^𝑙, 𝑅^𝑛×𝑅^𝑙, and 𝑅^𝑛, respectively.

The question arises as to whether the second-order fractional duality results developed in this paper hold for the following complex nondifferentiable minimax fractional problem:minimizeΨ(𝜉)=sup_𝜈∈𝑊Re𝑓(𝜉,𝜈)+𝑧^𝑇𝐵𝑧^1/2subjectto−𝑔(𝜉)∈𝑆,Re𝜉∈ℎ𝒞(𝜉,_2𝑛𝜈)−𝑧^𝑇𝐷𝑧^1/2,,(5.2) where 𝜉=(𝑧,𝑧),𝜈=(𝜔,𝜔) for 𝑧∈𝒞^𝑛,𝜔∈𝒞^𝑙,𝑓(⋅,⋅)∶𝒞^2𝑛×𝒞^2𝑙→𝒞 and ℎ(⋅,⋅)∶𝒞^2𝑛×𝒞^2𝑙→𝒞 are analytic with respect to 𝜔, 𝑊 ia a specified compact subset in 𝒞^2𝑙,𝑆 is a polyhedral cone in 𝒞^𝑚, and 𝑔∶𝒞^2𝑛→𝒞^𝑚 is analytic. Also 𝐵,𝐷∈𝒞^𝑛×𝑛 are positive semidefinite Hermitian matrices.

Acknowledgments

This paper is supported by Fast Track Project no. FT100023 of King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. The author is thankful to the referee for his/her valuable suggestions to improve the presentation of the paper.