# Solve The Optimization Problem

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For example, in Example \(\Page Index\), we are interested in maximizing the area of a rectangular garden.

Certainly, if we keep making the side lengths of the garden larger, the area will continue to become larger.

What size square should be cut out of each corner to get a box with the maximum volume?

Solution Step 1: Let \(x\) be the side length of the square to be removed from each corner (Figure \(\Page Index\)).

Intlinprog stopped because the objective value is within a gap tolerance of the optimal value, options. The intcon variables are integer within tolerance, options. The first step in the algorithm occurs as you place optimization expressions into the problem.

Branch and Bound: nodes total num int integer relative explored time (s) solution fval gap (%) 3627 0.64 2 2.154000e 03 2.593968e 01 5844 0.86 3 1.854000e 03 1.180593e 01 6204 0.91 3 1.854000e 03 1.455526e 00 6400 0.92 3 1.854000e 03 0.000000e 00 Optimal solution found. This conversion entails, for example, linear constraints having a matrix representation rather than an optimization variable expression.For nonlinear solver algorithms, see Unconstrained Nonlinear Optimization Algorithms and Constrained Nonlinear Optimization Algorithms.One common application of calculus is calculating the minimum or maximum value of a function.For example, companies often want to minimize production costs or maximize revenue.In manufacturing, it is often desirable to minimize the amount of material used to package a product with a certain volume.Local minimum found that satisfies the constraints. An object has an internal list of the variables used in its expressions.Optimization completed because the objective function is non-decreasing in feasible directions, to within the value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance. Branch and Bound: nodes total num int integer relative explored time (s) solution fval gap (%) 10000 1.13 0 - - 18188 1.85 1 2.906000e 03 4.509804e 01 22039 2.30 2 2.073000e 03 2.270974e 01 24105 2.51 3 1.854000e 03 9.973046e 00 24531 2.56 3 1.854000e 03 1.347709e 00 24701 2.58 3 1.854000e 03 0.000000e 00 Optimal solution found. Each variable has a linear index in the expression, and a size.Let’s look at how we can maximize the area of a rectangle subject to some constraint on the perimeter.Example \(\Page Index\): Maximizing the Area of a Garden A rectangular garden is to be constructed using a rock wall as one side of the garden and wire fencing for the other three sides (Figure \(\Page Index\)). Intlinprog stopped at the root node because the objective value is within a gap tolerance of the optimal value, options. The intcon variables are integer within tolerance, options. This table describes the exit flags for the or some other associated functions or objects. To correct these issues, try to scale the coefficient matrices, eliminate redundant linear constraints, or give tighter bounds on the variables.