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 solution:

Minimize subject to the constraint :

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

Find the solution:

Minimize 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 ():

Minimize over a region:

Plot it:

Find the minimum 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 variable:

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

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

Minimize a convex quadratic function subject to linear constraints:

Plot the region and minimizing point:

Minimize a convex quadratic function subject to a set of convex quadratic constraints:

Plot the region and the minimizing point:

Find the minimum distance between two convex regions:

Plot it:

Minimize such that is positive semidefinite:

Show the minimizer on a plot of the objective function:

Minimize the convex objective function such that is positive semidefinite and :

Plot the region and the minimizing point:

Minimize a convex objective function over a 4-norm unit disk:

Plot the region and the minimizing point:

Minimize the perimeter of a rectangle such that the area is 1 and 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:

Minimize 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 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:

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 and are convex functions:

Plot the region and the minimizing point:

Minimize a linear objective subject to nonlinear constraints:

Plot it:

Minimize a nonlinear objective subject to linear constraints:

Plot the objective and the minimizing point:

Minimize a nonlinear objective subject to nonlinear constraints:

Plot it:

This enforces a convergence criteria and :

This enforces a 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 solution along with the global minima:

Find the global minimum of a function containing multiple local minima using method "Couenne":

Plot the minimizing function and the global minimum solution:

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

Use method "Convex" or Except["Convex"] to specify if a convex method should be used or not:

Use method "Couenne" or Except["Couenne"] to choose or exclude the Couenne solver:

Steps taken by NMinimize 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 :

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 smallest ellipsoid parametrized as that encompasses a set of points in 3D by minimizing the volume:

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 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:

Minimize the objective :

The circles are contained in the square :

Compute the fraction of square covered by the circles:

Minimize 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 minimizing :

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:

Given a list of positive integers, partition the list into two non-overlapping subsets such that the difference between the sums of the two subsets is minimized. Define the list:

Define a binary variable that takes the values –1 or 1. If an element from the list belongs to subset 1, then the binary variable associated with it is given a value of –1. For subset 2, the associated binary variable is 1:

The objective is to minimize the difference of sums of the two subsets:

Find the subsets:

Given a collection of objects with different weights and two bags, find the objects with a maximum combined weight that can go into the two bags without exceeding the weight capacity of the bags. Define the number of objects and weights:

Use binary variables to decide if an object goes into bag 1 or 2, or neither:

The weight capacity of each bag is 5000:

The objective is to maximize the weights of the objects in the two bags:

Solve the problem:

The binary variable is associated with the objects that go into bag 1. The variable is the complement of . The weights in the bags are:

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 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 with distance of between points, where is the trajectory length being 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:

Find the number of teeth needed in the gears , respectively, to produce a specified gear ratio:

Each gear can have between and teeth:

For the given gear train configuration, the gear ratio is given by :

Find the number of teeth in each gear to achieve the gear ratio of :

Minimize the manufacturing cost of a pressure vessel. The vessel is a cylindrical shell of length and radius with a hemisphere cap at each end of the tube:

The volume constraint of the vessel is:

The shell has a thickness and the sphere cap has a thickness . The shell and cap thicknesses have specified upper and lower limits and must be greater than a fraction of the radius:

The thickness of the shell and cap have additional constraints:

The vessel dimension constraints are:

The shell and cap material has density . The material cost is:

The cost to weld the shell and cap is:

The total manufacturing cost is a total of welding and material costs:

Specify the constants:

Find the dimensions of the vessel that minimize the manufacturing cost:

Design a minimum-volume helical compression spring of winding diameter , spring wire diameter and number of spring coils subjected to an axial load:

The volume of the spring is:

The shear stress on the spring is dependent on the maximum allowable axial load and must be below the maximum allowable stress :

The deflection of the spring is defined as:

The spring free length must be less than the maximum allowable length :

The wire diameter must not exceed the specified minimum diameter:

The outside diameter of the spring must be less than the maximum specified diameter :

The winding diameter must be at least three times the wire diameter to avoid tightly wound springs:

The deflection under pre-load must be between a specified and :

The load induced from the deflection and pre-load must not exceed the maximum load :

The coil spring is to be manufactured from music wire spring steel ASTM A228 and the wire diameter must be chosen from a set of predetermined diameters:

The wire diameter is chosen from the predetermined set using binary variables:

Specify the constants:

Gather the constraints:

Solve the problem: