Find the minimum value of 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 minimum value:

Find the minimum value of subject to the constraint :

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

Find the minimum value:

Find the minimum value of 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 minimum value of over a region:

Find the minimum distance between two regions:

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

Find the minimum radius of a disk that contains the given three points:

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 variable:

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

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

Find the minimum value of a convex quadratic function subject to linear constraints:

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

Find the minimum distance between two convex regions:

Find the minimum value of such that is positive semidefinite:

Minimize convex objective function such that is positive semidefinite and :

Find the minimum value of a convex objective function over a 4-norm unit disk:

Find the minimum perimeter of a rectangle with area 1 such that the height is at most half the width:

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 the minimum value of 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 the minimum value of a linear objective subject to nonlinear constraints:

Find the minimum value of a nonlinear objective subject to linear constraints:

Find the minimum value of a nonlinear objective subject to nonlinear constraints:

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 objective function along with the global minimum value:

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

Steps taken by NMinValue 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 minimum distance between two 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 :

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

Find the minimum value of the radius subject to the constraints :

The minimal enclosing ball can be found efficiently using BoundingRegion:

Visualize the enclosing ball:

Find the smallest square that can contain circles of given radius for 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 bounds:

Find the Gaussian width parameter that minimizes the residual for an fit to nonlinear discrete data:

Fit the data using the basis :

The function will be approximated by :

Construct a parametric function that gives the residual:

Show the residual as a function of :

Find the scaling parameter that produces the minimum residual:

Find parameter such that the distance from the ellipse to the point (1,2) is as small as possible:

Show the distance as a function of :

Find the optimal parameter that maximizes the reciprocal of the distance:

Visualize the point with respect to the disk:

Find the minimum length of a path between the start and end points while avoiding circular obstacles:

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: