Find values of that minimize 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 minimizer point:

Find a point that minimizes subject to the constraint :

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

Find the minimizer point:

Find a point that minimizes 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 values of that minimize over a region:

Plot it:

Find points and that minimize the distance between two regions:

Plot it:

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

Plot it:

Find the disk of minimum 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 minimum 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 a point that minimizes a convex quadratic function subject to linear constraints:

Plot the region and minimizing point:

Find a point that minimizes a convex quadratic function subject to a set of convex quadratic constraints:

Plot the region and the minimizing point:

Find points and that minimize the distance between two convex regions:

Plot it:

Find a point that minimizes such that is positive semidefinite:

Show the minimizer on a plot of the objective function:

Find a point that minimizes the convex objective function such that is positive semidefinite and :

Plot the region and the minimizing point:

Find a point that minimizes a convex objective function over a 4-norm unit disk:

Plot the region and the minimizing point:

Find values of height and weight that minimize the perimeter of a rectangle with area 1 and :

This problem is log-convex and is solved by making a transformation {hExp[],wExp[ ]} and taking logarithms to get the convex problem:

Find a point that minimizes the quasi-convex function subject to inequality and norm constraints. The objective is quasi-convex because it is a product of a non-negative function and a non-positive function over the domain:

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:

Minimize 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:

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

Plot the region and the minimizing point:

Find a point that minimizes a linear objective subject to nonlinear constraints:

Plot it:

Find a point that minimizes a nonlinear objective subject to linear constraints:

Plot the objective and the minimizing point:

Find a point that minimizes 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 minimum point:

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

Steps taken by NArgMin in finding the minimum of the classic Rosenbrock function:

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

Find the closest points located on two different disks of radius 1 centered at and . Let be a point on disk 1. Let be a point on disk 2. The objective is to minimize subject to constraints :

Visualize the positions of the two points:

Find the radius and center of a minimal enclosing ball that encompasses a given region:

Minimize the radius subject to the constraints :

Visualize the enclosing ball:

The minimal enclosing ball can be found efficiently using BoundingRegion:

Find the values of and that minimize the volume of the ellipsoid parametrized as that encompasses a set of points in 3D:

For each point , the constraint must be satisfied:

Minimizing the volume is equivalent to minimizing :

Convert the parametrized ellipse into the explicit form :

A bounding ellipsoid, not necessarily minimum volume, can also be found using BoundingRegion:

Find the smallest square that can contain circles of given radii for and that do not overlap. Specify the number of circles and the radius of each circle:

If is the center of circle , then the objective is to minimize . The objective can be transformed so as to minimize and :

The circles must not overlap:

Collect the variables:

The circles are contained in the square . Find the bounding box and the centers:

Plot the circles and the bounding box:

Compute the fraction of square covered by the circles:

Find a point that minimizes subject to the constraints for a given matrix a and vector b:

Fit a cubic curve to discrete data such that the first and last points of the data lie on the curve:

Construct the matrix using DesignMatrix:

Define the constraint so that the first and last points must lie on the curve:

Find the coefficients by minimizing :

Compare fit with data:

Find a fit less sensitive to outliers to nonlinear discrete data by finding the value of that minimizes :

Fit the data using the bases . The interpolating function will be :

Find the solution:

Visualize the fit:

Compare the interpolating function with the reference function:

Find a line that separates two groups of points and :

For separation, set 1 must satisfy and set 2 must satisfy :

The objective is to minimize , which gives twice the thickness between and :

The separating line is:

Find a quadratic polynomial that separates two groups of 3D points and :

Construct the quadratic polynomial data matrices for the two sets using DesignMatrix:

For separation, set 1 must satisfy and set 2 must satisfy :

Find the separating polynomial by minimizing :

The polynomial separating the two groups of points is:

Plot the polynomial separating the two datasets:

Separate a given set of points into different groups. This is done by finding the centers for each group by minimizing , where is a given local kernel and is a given penalty parameter:

The kernel is a -nearest neighbor () function such that , else . For this problem, nearest neighbors are selected:

The objective is:

Find the group centers:

For each data point, there exists a corresponding center. Data belonging to the same group will have the same center value:

Extract and plot the grouped points:

Recover a corrupted image by finding an image that is closest under the total variation norm:

Create a corrupted image by randomly deleting 40% of the data points:

The objective is to minimize , where is the image data:

Assume that any nonzero data points are uncorrupted. For these positions, set :

Find the solution and show the restored image:

Find the positions of various cell towers and the range needed to serve clients located at :

Each cell tower consumes power proportional to its range, which is given by . The objective is to minimize the power consumption:

Let be a decision variable indicating that if client is covered by cell tower :

Each cell tower must be located such that its range covers some of the clients:

Each cell tower can cover multiple clients:

Each cell tower has a minimum and maximum coverage:

Collect all the variables:

Find the cell tower positions and their ranges:

Extract cell tower position and range:

Visualize the positions and ranges of the towers with respect to client locations:

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 :

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 a path through circular obstacles such that the distance between the start and end points is minimized:

The path is discretized into different points. The distance between these points must be less than where is the length to be minimized:

The points cannot be inside the circular objects:

The start and end points are known:

Collect the variables:

Minimize the length subject to the constraints:

Visualize the result: