SOME PROCEDURES FOR SOLVING SPECIAL MAX-MIN FRACTIONAL RANK-TWO REVERSE-CONVEX PROGRAMMING PROBLEMS

. In this paper, we suggest some procedures for solving two special classes of max-min fractional reverse-convex programs. We show that a special bilinear fractional max-min reverse-convex program can be solved by a linear reverse-convex programming problem. For a linear fractional max-min reverse-convex program, possessing two re-verse-convex sets, we propose a parametrical method. The particularity of this procedure is the fact that the max-min optimal solution of the original problem is obtained by solving at each iteration two linear reverse-convex programs with a rank-two monotonicity property


INTRODUCTION
In this paper, we propose some methods for solving two special classes of max-min fractional reverse-convex programs.
Let X ⊆ R n , Y ⊆ R m be given convex sets.Let T be a reverse-convex set in R n (i.e. the complement of a convex set in R n ) and S be a reverse-convex set in R m and h : X × Y → R. The general max-min reverse-convex programming consists in finding an optimal max-min solution for:

PRC. max
x∈X∩T min y∈Y ∩S h(x, y).
We recall (see, e.g.[14,17]) that a point (x , y ) ∈ (X ∩ T ) × Y is said to be an optimal max-min solution for PRC problem, if the following two conditions hold: (i) h(x , y ) = max In the particular case, when PRC is a simple maximization problem only (i.e.Y has a single element, X is a polyhedral set and T is a reverse-convex set defined by a convex or quasi-convex constraint which has a rank-two monotonicity property), we recall that procedures for solving the maximum (or minimum) reverse-convex programming problems were given, for instance, in the papers [4], [7], [8], [10], [18], [6].Duality aspects of the reverse-convex programming can be found in the paper by Penot [9].For the max-min reverseconvex programming we mention the refs.[5], [6].
In section 2, for a special fractional type of the objective function h and for some particular cases of the sets X, Y, T and S, we obtain a particular case of problem PRC of bilinear fractional form, for which we propose, in Section 3, a method of finding optimal max-min solutions by reducing these problems to the particular case when X is a polyhedral set and T is a reverseconvex set defined by a convex or quasi-convex constraint that has a rank-two monotonicity property.These auxiliary problems can be solved by Kuno-Yamamoto procedure [8].
In section 4, we consider a max-min linear fractional reverse-convex programming problem, having two reverse-convex sets T and S, and for which we propose a parametric procedure.Some concluding remarks are made in the last section.

BILINEAR FRACTIONAL MAX-MIN REVERSE-CONVEX PROGRAM PFM
Next we consider the following bilinear fractional max-min reverse-convex program (see, [6]) in which the reverse set S = R m : PFM.Find (1) The sets X and Y are defined by where is defined by a function f : R n → R, which is continuous, strictly quasiconcave and has rank-two monotonicity on an open convex set X 0 ⊆ R n , which includes the set X.
We recall that f is said to be strictly quasiconcave on X 0 if for each x, y ∈ X 0 with f (x) = f (y) we have f ((1 − t)x + ty) > min{f (x), f (y)}, for any t ∈ (0, 1).Definition 1. [8] The function f possess a rank-two monotonicity on X 0 with respect to linearly independent vectors λ 1 , λ 2 ∈ R n , if for any points x , x ∈ X 0 , the inequality We have the following representation for a function f possessing a rank-two monotonicity on X 0 with respect to linearly independent vectors λ 1 , λ 2 ∈ R n .Lemma 2. [8] If the function f possess a rank-two monotonicity on X 0 with respect to linearly independent vectors λ 1 , λ 2 ∈ R n , then there exists a function g : R 2 → R, which is continuous and strictly quasiconcave on Γ 0 = {(λ 1 x, λ 2 x)|x ∈ X 0 } and satisfies the following two conditions: Concerning the problem PFM, we make the following assumptions: A1.Y is a bounded non-empty set, A2. the function f is continuous, strictly quasi-concave and has rank-two monotonicity on the open convex set X 0 , A3. max{α i y + β i |i ∈ I} > 0, ∀y ∈ Y.

LINEAR REVERSE-CONVEX PROGRAMMING APPROACH
We proposed in [6] a procedure for solving problem PFM based on the Charnes-Cooper [1] variable change.
Thus, if we perform in the problem ( 1)-( 4), the Charnes-Cooper variable change v = ty, t ≥ 0, v ∈ R s , (see, also Schaible [11], Stancu-Minasian [12] and Tigan [14], [16]) it follows by assumptions A1 and A3 that problem PFM is equivalent with: where the set Y is defined by: We can show, by assumptions A1 and A3, that V = V and that for any optimal max-min solution (x * , y * ) of problem PFM there exists an optimal max-min solution (x * , v * , t * ) of problem PA such that y * = v * t * and conversely.By Golstein [3] (see, also T ¸igan and Stancu-Minasian [17]), the constraint (7) in Problem PA can be rewritten as the following maximum bilinear constraint: Let us denote By linear programming duality, for any (v, t) ∈ Y , we have 10)-( 12), it follows that the inequation ( 9) can be expressed by the following system: Therefore, problem PA can be reduced to the following max-min bilinear reverse-convex program PMM1.Find where the set Y is defined by: From ( 13)-( 16), by linear programming duality, for any x ∈ X ∩ T, problem PMM1 can be transformed into the following linear reverse-convex program: In problem PML, we denote by Ω ∈ R r×s the matrix having the rows α i (i ∈ I), and by Λ the vector Λ = (β 1 , ..., β r ) T ∈ R r .
Therefore, we proved the following theorem: Theorem 3. If the problem PFM satisfies the assumptions A1-A3, then problem PFM can be solved by solving the linear reverse-convex program with a rank-two monotonicity PML.
In order to solve problem PFM, a procedure similar to Algorithm 1 can be used, by replacing in step1 the problem PLC by the linear reverse-convex programming problem PML.

Algorithm 1. Step 1. Solve the linear reverse-convex programs with a rank-two monotonicity PML.
If V < ∞ and the feasible set X of PML is non-empty, let x * be the corresponding component of an optimal solution of problem PML.
) is an optimal solution of maxmin problem PFM, where y * is an optimal solution of the generalized linearfractional program PFA. ii We make the remark that auxiliary linear reverse-convex programming problem PML in Algorithm 1 is simpler than the auxiliary linear reverseconvex programming problem in the algorithm proposed for this problem in [6].Indeed, problem PML has only s+r linear constraints the auxiliary problem in [6] posses (s + 2)r linear constraints.Therefore, Algorithm 1 seems to be more efficient than algorithm proposed for this problem in [6].

MAX-MIN LINEAR FRACTIONAL REVERSE-CONVEX PROGRAMMING WITH TWO SEPARATE REVERSE-CONVEX FEASIBLE SETS
In this section, we consider the following max-min linear fractional program GLF.Find max where verifying the condition In problem GLF, X and S are defined by For solving problem GLF we can use a parametric procedure (see, [2], [13], [15]), by which an approximate optimal solution could be found by solving a sequence of the auxiliary reverse-convex programs each of them having only one reverse-convex constraint.
Algorithm 2. Let ε > 0 be a given positive real number, representing a level of approximation to be attain by algorithm.
1. Find a point x 0 ∈ X ∩ T and a point y 0 ∈ Y ∩ S and set k := 0.

Take t
But the max-min program (22) can be transformed into the following two linear reverse-convex programs PL1.Find (23) max Let x k+1 , y k+1 be an optimal solution of the linear reverse-convex program (23)-( 24) and ( 25)-(26), respectively.Obviously, we have ) is an approximate optimal solution of problem GLF.
ii) If F (t k ) > ε, then take k:=k+1 and go to the step 2.

CONCLUSIONS
In this paper we consider two fractional max-min reverse-convex programming problems.
Firstly, we give a new procedure for solving a particular class of maxmin bilinear fractional reverse-convex programming problems.The particularity of this procedure is the fact that the max-min optimal solution of the original problem is obtained by solving a single linear reverse-convex program with a rank-two monotonicity with an algorithm proposed by Kuno and Yamamoto [8].
Secondly, we consider a parametric procedure for solving a particular class of max-min linear fractional reverse-convex programming problems, possessing two reverse-convex feasible sets.The particularity of this procedure is the fact that the max-min optimal solution of the original problem is obtained by solving at each iteration two linear reverse-convex programs with a rank-two monotonicity with the algorithm of Kuno and Yamamoto.
are given matrices, vectors and real constants respectively.The reverse-convex set (20) T = {x ∈ X 0 |f (x) ≤ 0}, is defined by a function f : R n → R, which is continuous, strictly quasiconcave and has rank-two monotonicity on an open convex set X 0 ⊆ R n , which includes the set X and the reverse-convex set (21) S = {y ∈ Y 0 |f 1 (y) ≤ 0}, is defined by a function f 1 : R m → R, which is continuous, strictly quasiconcave and has rank-two monotonicity on an open convex set Y 0 ⊆ R m , which includes the set Y .