Maximize subject to constraints :

Several linear inequality constraints can be expressed with VectorGreaterEqual:

Use v>= or \[VectorGreaterEqual] to enter the vector inequality sign :

An equivalent form using scalar inequalities:

Use a vector variable :

The inequality may not be the same as due to possible threading in :

To avoid unintended threading in , use Inactive[Plus]:

Use constant parameter equations to avoid unintended threading in :

VectorGreaterEqual represents a conic inequality with respect to the "NonNegativeCone":

To explicitly specify the dimension of the cone, use {"NonNegativeCone",n}:

Find the solution:

Maximize subject to the constraint :

Specify the constraint using a conic inequality with "NormCone":

Find the solution:

Maximize the function subject to the constraint :

Use Indexed to access components of a vector variable, e.g. :

Use Vectors[n,dom] to specify the dimension and domain of a vector variable when it is ambiguous:

Specify non-negative constraints using NonNegativeReals ():

An equivalent form using vector inequality :

Specify non-positive constraints using NonPositiveReals ():

An equivalent form using vector inequalities:

Or constraints can be specified:

Specify integer domain constraints using Integers:

Specify integer domain constraints on vector variables using Vectors[n,Integers]:

Specify non-negative integer domain constraints using NonNegativeIntegers ():

Specify non-positive integer domain constraints using NonPositiveIntegers ():

Find the maximum value of a linear function of a vector at in with :

Find the maximum value over a region:

Find the maximum possible distance between two points constrained to be in two different regions:

With linear objectives and constraints, when a maximum is found it is global:

The constraints can be equality and inequality constraints:

Use Equal to express several equality constraints at once:

An equivalent form using several scalar equalities:

Use VectorLessEqual to express several LessEqual inequality constraints at once:

Use v<= to enter the vector inequality in a compact form:

An equivalent form using scalar inequalities:

Use Interval to specify bounds on a variable:

Use "NonNegativeCone" to specify linear functions of the form :

Use v>= to enter the vector inequality in a compact form:

Find the maximum value of a concave quadratic function subject to a set of convex quadratic constraints:

Plot the region:

Find the maximum value of such that is positive semidefinite:

Show the plot of the objective function:

Find the maximum value of a concave objective function such that is positive semidefinite and :

Plot the region and the maximizing point:

Maximize the quasi-concave function subject to inequality and norm constraints. The objective is quasi-concave because it is a product of two non-negative affine functions:

The maximization is solved by minimizing the negative of the objective, , that is quasi-convex. Quasi-convex problems can be solved as parametric convex optimization problems for the parameter :

Plot the objective as a function of the level-set :

For a level-set value between the interval , the smallest objective is found:

The problem becomes infeasible when the level-set value is increased:

Maximize subject to the constraint . The objective is not convex but can be represented by a difference of convex function where and are convex functions:

Plot the region and the minimizing point:

Maximize subject to the constraints . The constraint is not convex but can be represented by a difference of convex constraint where and are convex functions:

Plot the region and the minimizing point:

Maximize a linear objective subject to nonlinear constraints:

Plot it:

Maximize a nonlinear objective subject to linear constraints:

Plot the objective and the maximizing point:

Maximize a nonlinear objective subject to nonlinear constraints:

Plot it:

This enforces convergence criteria and :

This enforces convergence criteria and , which is not achievable with the default machine-precision computation:

Setting a high WorkingPrecision makes the process convergent:

Record all the points evaluated during the solution process of a function with a ring of minima:

Plot all the visited points that are close in objective function value to the final solution:

Some methods may give suboptimal results for certain problems:

The automatically chosen method gives the optimal solution for this problem:

The automatic method choice for this nonconvex problem is method "Couenne":

Plot the objective function along with the global maximum value:

Use method "NelderMead" for problems with many variables when speed is essential:

Steps taken by NMaxValue in finding the maximum of a function:

With the working precision set to , by default AccuracyGoal and PrecisionGoal are set to :

Find the maximum such that the triangle and ellipse still intersect:

Plot the resulting regions:

Find the largest radius for non-overlapping circles and their centers that can be contained in a square. The box constraint can be represented as :

The circles must not overlap:

Collect the variables:

Find the maximum radius :

Find the maximum return on investment possible by allocating $250,000 of capital to purchase two stocks and a bond. Let be the amount to invest in the two stocks and let be the amount to invest in the bond:

The amount invested in the utilities stock cannot be more than $40,000:

The amount invested in the bond must be at least $70,000:

The total amount invested in the two stocks must be at least half the total amount invested:

The stocks pay an annual dividend of 9% and 4%, respectively. The bond pays a dividend of 5%. The total return on investment is:

The cost of executing the transactions is and must not exceed $1000:

The maximum profit attainable with the specified constraints is:

Find the value for the parameter associated with ellipse such that the maximum value of is minimized:

Show the distance as a function of :

Find optimal parameter that minimizes the maximum value of the objective: