Minimize subject to the constraints and :

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 subject to the positive semidefinite matrix constraint :

Find the solution:

Use a vector variable and Indexed[x,i] to specify individual components:

Use Vectors[n,Reals] to specify the dimension of a vector variable when it may be ambiguous:

Specify non-negative constraints using NonNegativeReals ():

An equivalent form using vector inequality :

Maximize the area of a rectangle with perimeter at most 1 and height at most half the width:

When and are positive, the problem can be solved by GeometricOptimization methods:

Using method GeometricOptimization implicitly assumes positivity:

Specify integer variables 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 ():

Specify complex variables using Complexes:

Minimize a real objective with complex variables and complex constraints :

Let . Expanding out the constraints into real components gives:

Solve the problem with real-valued objective and complex variables and constraints:

Solve the same problem with real variables and constraints:

Use a quadratic objective with Hermitian matrix and real-valued variables:

Use objective (1/2)Inactive[Dot][Conjugate[x],q,x] with a Hermitian matrix and complex variables:

Use a quadratic constraint with Hermitian matrix and real-valued variables:

Use constraint (1/2)Inactive[Dot][Conjugate[x],q,x]d with a Hermitian matrix and complex variables:

Find the Hermitian matrix with minimum 2-norm (largest singular value) such that the matrix is positive semidefinite:

The minimum for the largest singular value is:

Use a linear matrix inequality constraint with Hermitian or real symmetric matrices:

The variables in linear matrix inequalities need to be real for the sum to remain Hermitian:

Minimize over the intersection of a triangle and a disk :

Get the primal minimizer as a vector:

Get the minimal value:

Plot the solution:

Minimize subject to and :

The dual problem is to maximize subject to :

The primal minimum value and the dual maximum value coincide because of strong duality:

That is the same as having a duality gap of zero. In general, at optimal points:

Get the dual maximum value and dual maximizer directly using solution properties:

The "DualMaximizer" can be obtained with:

The dual maximizer vector partitions match the number and dimensions of the dual cones:

To get the dual format for a particular problem-type solver, specify it as a method option:

"SCS" is a splitting conic solver method:

"CSDP" is an interior point method for semidefinite problems:

"DSDP" is an alternative interior point method for semidefinite problems:

"IPOPT" is an interior point method for nonlinear problems:

Different methods have different default tolerances, which affects the accuracy and precision:

Compute exact and approximate solutions:

"SCS" has a default tolerance of :

"CSDP", "DSDP" and "IPOPT" have default tolerances of :

When method "SCS" is specified, it is called with the SCS library default tolerance of 10^-3:

With default options, this problem is solved by method "SCS" with tolerance 10^-6:

Use methods "CSDP" or "DSDP" for constraints that are converted to semidefinite constraints:

Solve the problem using method "CSDP":

Solve the problem using method "DSDP":

Use method "IPOPT" to obtain accurate solutions when "CSDP" and "DSDP" are not applicable:

"IPOPT" produces more accurate results than "SCS", but is typically much slower:

Compare timing with method "SCS":

The default value of the option PerformanceGoal is $PerformanceGoal:

Use PerformanceGoal"Quality" to get a more accurate result:

Use PerformanceGoal"Speed" to get a result faster, but at the cost of quality:

Compare the timings:

The "Speed" goal gives a less accurate result:

A smaller Tolerance setting gives a more precise result:

Compute the exact minimum value with Minimize:

Compute the error in the minimum value with different Tolerance settings:

Visualize the change in minimum value error with respect to tolerance:

A smaller Tolerance setting gives a more precise answer, but may take longer to compute:

The tighter tolerance gives a more precise answer:

The default working precision is MachinePrecision:

Using WorkingPrecisionInfinity will give an exact solution if possible:

WorkingPrecision other than MachinePrecision and ∞ will try to use a method with extended precision support:

Using WorkingPrecisionAutomatic will try to use the precision of the input problem:

Solve a problem with a quadratic objective using 24-digit precision:

There is currently no method that solves problems with quadratic objectives using exact arithmetic. When the requested precision is not supported, the computation uses machine numbers:

Maximize subject to . Solve a maximization problem by negating the objective function:

Negate the primal minimum value to get the corresponding maximal value:

Minimize subject to . Since the constraint is not convex, use a semidefinite constraint to make the convexity explicit:

A matrix is positive semidefinite if and only if the determinants of all upper-left submatrices are non-negative:

Find the solution:

Minimize subject to , assuming when . Using the auxiliary variable , the objective is to minimize such that :

Check that implies :

A Schur complement condition says that if , a block matrix iff . Therefore, iff . Use Inactive[Plus] for constructing the constraints to avoid threading:

Minimize over an ellipse centered at :

The epigraph transformation can be used to construct a problem with a linear objective and additional variable and constraint:

In this form, the problem can be solved directly with ConicOptimization:

Minimize , where is a nondecreasing function, by instead minimizing . The primal minimizer will remain the same for both problems. Consider minimizing subject to :

The minimum value for can be obtained by applying to the minimum value of :

ConvexOptimization will automatically do this transformation:

Find that minimizes the largest eigenvalue of a symmetric matrix that depends linearly on the decision variables , . The problem can be formulated as a linear matrix inequality since is equivalent to , where is the eigenvalue of . Define the linear matrix function :

A real symmetric matrix can be diagonalized with an orthogonal matrix so . Hence iff . Since any , taking , , hence iff . Numerically simulate to show that these formulations are equivalent:

The resulting problem:

Run a Monte Carlo simulation to check the plausibility of the result:

Find that maximizes the smallest eigenvalue of a symmetric matrix that depends linearly on the decision variables . Define the linear matrix function :

The problem can be formulated as linear matrix inequality, since is equivalent to where is the eigenvalue of . To maximize , minimize :

Run a Monte Carlo simulation to check the plausibility of the result:

Find that minimizes the difference between the largest and the smallest eigenvalues of a symmetric matrix that depends linearly on the decision variables . Define the linear matrix function :

The problem can be formulated as a linear matrix inequality, since is equivalent to , where is the eigenvalue of . Solve the resulting problem:

In this case, the minimum and maximum eigenvalues coincide and the difference is 0:

Minimize the largest (by absolute value) eigenvalue of a symmetric matrix that depends linearly on the decision variables :

The largest eigenvalue satisfies The largest (by absolute value) negative eigenvalue of is the largest eigenvalue of and satisfies :

Find that minimizes the largest singular value of a symmetric matrix that depends linearly on the decision variables :

The largest singular value of is the square root of the largest eigenvalue of , and from a preceding example it satisfies , or equivalently :

Plot the result:

For quadratic sets , which include ellipsoids, quadratic cones and paraboloids, determine whether , where are symmetric matrices, are vectors and scalars:

Assuming that the sets are full dimensional, the S-procedure says that iff there exists some non-negative number such that Visually see that there exists a non-negative :

Use 0 for an objective function since feasibility is a concern. Since λ≥0, it follows that :

Minimize the length of the diagonal of a rectangle of area 4 such that the width plus three times the height is less than 7:

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:

The distance between the points is:

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:

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 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 . Taking and negating the objective, the transformed objective is :

Using auxiliary variable , the transformed objective is subject to the constraint :

Visualize the location of the center:

Test whether an ellipsoid is a subset of another ellipsoid of the form :

Using the S-procedure, it can be shown that ellipse 2 is a subset of ellipse 1 iff :

Check if the condition is satisfied:

Convert the ellipsoids into explicit form and confirm that ellipse 2 is within ellipse 1:

Move ellipsoid 2 such that it overlaps with ellipsoid 1:

A test now shows that the problem is infeasible, indicating that ellipsoid 2 is not a subset of ellipsoid 1:

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:

Minimizing the area is equivalent to minimizing , which is equivalent to minimizing :

Convert the parametrized ellipse into the explicit form as :

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 , which 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:

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 an regularized fit to complex data by minimizing for a complex :

Construct the matrix using DesignMatrix, for the basis :

Find the coefficients :

Let be the fit defined as a function of the real and imaginary components of :

Visualize the result for the real component of :

Visualize the results for the imaginary component of :

Represent a given polynomial in terms of the sum-of-squares polynomial :

The objective is to find such that , where is a vector of monomials:

Construct the symmetric matrix :

Find the polynomial coefficients of and and make sure they are equal:

Find the elements of :

The quadratic term , where is a lower-triangular matrix obtained from the Cholesky decomposition of :

Compare the sum-of-squares polynomial to the given polynomial:

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:

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 , 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:

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: