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198 Andrey V. OrlovStep 4. Beginning from (ū s , ¯v s ), by X-procedure or Y-procedure obtain δ k -critical point (ˆx s , ŷ s ) ∈D in (P).Step 5. Find δ k -solution (x s 0 , ys 0 ), f(xs 0 , ys 0 ) = γ + ζ k, of level problemStep 6. Compute〈∇ x f(x s 0 , ys 0 ), ˆxs − x s 0 〉 + 〈∇ yf(x s 0 , ys 0 ), ŷs − y0 s〉 + δ k ≥≥ sup{〈∇ x f(x, y), ˆx s − x〉 + 〈∇ y f(x, y), ŷ s − y〉 : f(x, y) = γ + ζ k }. (11)x,yη k (γ) = γ − g(ˆx s , ŷ s ) + 〈∇ x f(x s 0 , ys 0 ), ˆxs − x s 0 〉 + 〈∇ yf(x s 0 , ys 0 ), ŷs − y0 s 〉. (12)Step 7. If η k (γ) ≤ 0, s < N, then let s := s + 1 and go to step 2.Step 8. If η k (γ) ≤ 0, s = N, then let γ := γ + △γ, s := p and go to step 2.Step 9. If η k (γ) > 0, then let (¯x k+1 , ȳ k+1 ) := (ˆx s , ŷ s ), k := k + 1, s := s + 1, p := s and go tostep 1.Step 10. If s = N, η k (γ) ≤ 0, ∀γ ∈ [γ − , γ + ] then Stop.Let us explain some steps and parameters of above algorithm.On step 2 we construct the level surface approximation by the set of vectors Dir. On step 3we examine the point of level surface approximation on applicability by the Global OplimalityConditions inequality (see [2]). Scalar ν is a first parameter of algorithm. By this parameter wecan change accuracy of inequality on step 3. On step 4 we implement additional local searchinstead of linearized problem solving (see [2]).On step 6 we compute the quality assessment of algorithm’s iteration. On steps 7–10 wecheck on stopping criterion and perform returns on internal (on points of level surface approximationand on parts of segment [γ − , γ + ]) and external (on critical points) cycles. Scalarq is a second parameter of algorithm. By this parameter we can change number of abovementioned segment parts.The testing of the developed algorithm performed on the special disjoint bilinear program.This program is equivalent to finding Nash equilibrium point in bimatrix games [4].AcknowledgmentsThe author wish to thank prof. A.S. Strekalovsky for their encouragement and support.References[1] Vasil’ev, F.P. (1988). Numerical methods of extremal problems solving. Nauka, Moscow (in russian).[2] Strekalovsky, A.S. (2003). Elements of Nonconvex Optimization. Nauka, Novosibirsk (in russian).[3] Mukhamediev, B.M. (1978). The solution of bilinear problems and finding the equilibrium situations in bimatrixgames. Computational mathematics and Mathematical Physics, Vol.18, 60–66.[4] Orlov, A.V., Strekalovsky, A.S. (2004). Seeking the Equilibrium Situations in Bimatrix Games. Automation andremote control, Vol.65, 204–217.[5] Mangasaryan, O.L. (1993). Bilinear Separation of Two Sets in n-Space. Computational Optimization and Applications,No.2, 207–227.[6] Floudas, C.A., Visweswaran, V. (1995). Quadratic optimization. In Handbook of Global Optimization/Ed. byHorst. R., Pardalos P. Dordrecht: Kluwer Academic Publishers, 224–228.