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:

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 ():

Or constraints can be specified:

Maximize over a region:

Plot it:

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

Plot it:

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

Plot it:

Find the circle of maximum radius that contains the given three points:

Plot it:

Using Circumsphere gives the same result directly:

Use to specify that is a vector in with :

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 variable:

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

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

Minimize such that is positive semidefinite:

Show the maximizer on a plot of the objective function:

Minimize the 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 a parametric convex optimization problem 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:

Plot the solution and show the global maxima:

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

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

Find the half-lengths of the principal axes that maximize the volume of an ellipsoid with a surface area of at most 1:

The surface area can be approximated by:

Maximize the volume area by minimizing its reciprocal:

This is the sphere. Including additional constraints on the axes lengths changes this:

Plot the resulting ellipsoid:

Find the analytic center of a convex polygon. The analytic center is a point that maximizes the product of distances to the constraints:

Each segment of the convex polygon can be represented as intersections of half-planes . Extract the linear inequalities:

The objective is to maximize . Since the logarithm function is monotonic, this is equivalent to maximizing :

Solve the unconstrained problem:

Visualize the location of the center:

Find the maximum-area ellipse parametrized as that can be fitted into a convex polygon:

Each segment of the convex polygon can be represented as intersections of half-planes . Extract the linear inequalities:

Applying the parametrization to the half-planes gives . The term . Thus, the constraints are:

Maximizing the area is equivalent to maximizing :

Convert the parametrized ellipse into the explicit form as :

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 and the circle centers :

Plot the circles and the bounding box:

Compute the fraction of square covered by the circles:

Find the distribution of capital to invest in six stocks to maximize return while minimizing risk:

The return is given by , where is a vector of the expected return value of each individual stock:

The risk is given by ; is a risk-aversion parameter and :

The objective is to maximize return while minimizing risk for a specified risk-aversion parameter:

The effect on market prices of stocks due to the buying and selling of stocks is modeled by , which is modeled by a power cone using the epigraph transformation:

The weights must all be greater than 0 and the weights plus market impact costs must add to 1:

Compute the returns and corresponding risk for a range of risk-aversion parameters:

The optimal over a range of gives an upper-bound envelope on the tradeoff between return and risk:

Compute the weights for a specified number of risk-aversion parameters:

By accounting for the market costs, a diversified portfolio can be obtained for low risk aversion, but when the risk aversion is high, the market impact cost dominates, due to purchasing a less diversified stock:

Find the allocation of a maximum of $250,000 of capital to purchase two stocks and a bond such that the return on investment is maximized. 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 with purchasing the stocks and bonds and executing the transactions is :

The optimal amount to be invested can be found by maximizing the profit while minimizing cost:

The total amount invested and the annual dividends received from the investments are:

Find the optimal combination of investments that yields maximum profit. The net present value and the costs associated with each investment are:

Let be a decision variable such that if investment is selected. The objective is to maximize profits while minimizing costs:

There is a maximum $14,000 available for investing:

Solve the maximization problem to get the optimal combination of investments:

Find the path that a salesman should take through cities such that each city is only visited once, travel cost savings are maximized and distance traveled is minimized. Generate the locations:

Let be the distance between city and city . Let be a decision variable such that if , the path goes from city to city :

The travel budget to go from one city to another is $15. If two cities are within 50 miles, the salesman pays $5; otherwise, he pays a $10 flat rate. The total savings are:

The objective is to maximize savings while minimizing distance:

The salesman can arrive from exactly one city and can depart to exactly one other city:

The salesman cannot arrive at one city and depart to the same city:

The salesman must travel to all the locations in a single tour:

The decision variable is a binary variable and the dummy variable is :

Find the path that minimizes the distance:

Extract the path:

The distance traveled is:

The total savings is: