# Methods of Optimization (final work)

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Final Examination
1. Factory for the production of ice cream can produce five kinds of ice cream. In the production of ice cream are two different kinds of raw materials: milk and excipients, which are known stocks.
It is known as the unit cost of raw materials and product prices. Required to construct a production plan that maximizes revenue.

2.Find and depict the Cartesian field of convexity and concavity of the function f {xty) = (x-1) 3 - bhu + v3. Convex if constructed area?

lead to the standard form. Show admissible set and the line level of the objective function; to solve the problem graphically. Check whether the conditions of the existence theorem of Weierstrass
solutions. Figure verify the conditions of Kuhn-Tucker at the corner points of the feasible set (that is, points at which the number of active constraints is not less than the number of variables) and the points of tangency of the level curve of the objective function with the boundaries of the permissible area. Find the point at which the Kuhn-Tucker conditions are met, and to determine which of the constraints are active at such points.
Prescribe the conditions of Kuhn-Tucker points to found and calculated values \u200b\u200bof the dual variables. Make a reasonable conclusion about the presence or absence of local (global) maximum in all the points.

4.Rassmotret goal programming problem, in which the set of feasible solutions is given by the inequalities xi + 2x2 <4, 4xx + x2 4 and X12> 0, the criteria set
relations z2 = 2x1 + x2, z2 = 2x2, and the target point coincides with the ideal point z *, the deviation from that given by the function /? (z, z *) = max {(Zj * -Zj), (z2 * -z2) }. Find and represent the set of criteria vectors Z, it paretovu border P (Z) and an ideal point z *. Show the level lines of /? (Z, z *). Graphically solve the problem of finding an achievable point (z'i, z 2), which gives a minimum deviation from the ideal points; Analytical record the problem of minimizing deviations from the ideal point in the form of a linear programming problem.

5. Consider the problem of maximizing twocriterial
zl = F1 (x) = 2xj + 5x2 + 4x3? max, z2 = F2 (x) = x2 + -5hh - 4x3? max
on the set of admissible solutions X and. E3
2? + X | + (Xj +1) 2 <1 x /? Oh, x2? Oh, xj? 0.
Find Pareto-efficient solution that maximizes linear convolution of criteria
^ (Z1, z2) = 0,6z1 + 0,4z2.
Check whether the problem arises for nonlinear programming conditions of the theorem of Weierstrass and whether it is a convex programming problem. Check the use of the Kuhn-Tucker in this problem. Write out and check the conditions of Kuhn-Tucker in the gradient form for different sets of active constraints. Find the solution of the nonlinear programming problem. Write out the function of Lagrange and Kuhn-Tucker through the Lagrangian; to verify the conditions of Kuhn-Tucker found in the decision.

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