Question

Problem 8-01 (Algorithmic) The purpose of this exercise is to provide practice using the LINGO or Excel solvers. Find the values of X and Y that minimize the function Min x2 - 8X + y? + 6Y+ 25 Do not assume nonnegativity of the X and Y variables. Recall that by default LINGO assumes nonnegative variables. In order to allow the variables to take on negative values you can add FREE (X); @ FREE (Y); Alternatively, if you want LINGO to allow for negative values by default, in the LINGO menu select Options and then click General Solver, and then uncheck the Variables assumed nonetegative tab. To allow for negative values in Excel Solver, make sure that the Make Unconstrained Variables Non-Negative box is not checked in the Solver Parameters dialog box. Round your answers to the nearest whole number. If negative answer is required, enter the minus sign before the number. Optimal solution is X for an optimal solution value of 0 Y =

Answer 1

Setup of formula:

=A2^2-8*A2+B2^2+6*B2+25 B2 A B F F f 1 1-A2 2-8*A2+B2^2+6*B2+25 4 X co .. st

Next we setup the Solver for minimization (Note the make unconstrained variables non-negative box is unchecked)

Get External Data Connections Solver Parameters =A2^2-8* A2+B2^2+6*B2+25 C2 SCS2 Set Objective: C D E P 1 x f O Value Of: 25995 To: O Min OMax 2 1 By Changing Variable Cells: SA$2:SB$2 4 Subject to the Constraints: 6 Add 7 8 Change 9 10 Delete 11 12 Reset All 13 14 Load/Save 15 Make Unconstrained Variables Non-Negative 16 Select a Solving Method: GRG Nonlinear 17 Options 18 Solving Method 19 Select the GRG Nonlinear engine for Solver Problems that are smooth nohlhear. Select Problems, an e LP 20 select volutionary engine problems that are non-smooth. 21 22 23 Help Solve Close Sheet3 Pollard Rho Sh Cavley table Sheet8

And the solution we get is:

=A2^2-8*A2+B2^2+6* B2+25 C2 A B F Solver Results X f Solver found a solution. All Constraints and optimality 2 4 -3 0 conditions are satisfied. Reports 3 Answer 4 O Keep Solver Solution Sensitivity 5 Limits ORestore Original Values 7 8 Return to Solver Parameters Dialog Outline Reports 10 Cancel OK Save Scenario... 11 12 Solver found a solution. All Constraints and optimality conditions are satisfied 13 When the GRG engine is used, Solver has found at least a local optimal solution. When Simplex LP is used, this means Solver has found a global optimal solution. 14 15 16 17

Thus, the minimum value is achieved at
T, y) = (4, -3)
and the minimum value is

$\blacksquare$

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