Minimize subject to constraints :

Minimize when the matrix is positive semidefinite:

Find the solution:

Minimize subject to the linear matrix inequality constraint :

The left-hand side of the constraint can be given in evaluated form:

Express the problem with the objective vector and constraint matrices:

Use a vector variable :

Use a vector variable and Inactive[Plus] to avoid unintended threading:

Use a vector variable and parameter equations to avoid unintended threading:

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

Use Vectors[n] to specify the dimension of a vector variable when it is ambiguous:

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 and vector inequality:

Specify non-negative constraints using NonNegativeReals ():

An equivalent form using vector inequalities:

Second-order cone constraints of the form can be used:

"NormCone" constraints of the form can be used:

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 ():

Specify complex variables using Complexes:

In linear matrix inequalities, the constraint matrices can be Hermitian or real symmetric:

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

Minimize the function subject to the constraint :

Get the primal minimizer as a vector:

Get the minimal value:

Extract the objective vector:

Extract the constraint matrices:

Use the extracted objective vector and constraint matrices for direct input:

The slack for an inequality at the minimizer is given by :

Extract the minimizer and constraint matrices:

Verify that the slack matrix satisfies :

Minimize subject to :

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:

Construct the dual problem using constraint matrices extracted from the primal problem:

Extract the objective vector and constraint matrices:

The dual problem is to maximize subject to :

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

The "DualMaximumValue" is:

The "DualMaximizer" can be obtained with:

Use "ConstraintSensitivity" to find the change in optimal value due to constraint perturbations:

The sensitivity is a matrix:

Consider new constraints where is the perturbation:

The approximate new optimal value is:

Compare to directly solving the perturbed problem:

The optimal value changes according to the signs of the sensitivity matrix elements:

At negative sensitivity element position, a positive perturbation will decrease the optimal value:

At positive sensitivity element position, a positive perturbation will increase the optimal value:

Express the perturbed constraints symbolically using Sylvester's criterion for semidefiniteness:

With this form, the minimum value can be found exactly as a function of the parameters around 0:

Make a symmetric matrix with the derivatives of the minimum with respect to the parameters:

This is the sensitivity to perturbation given by the "ConstraintSensitivity" property:

The constraint sensitivity can also be obtained as the negative of the dual maximizer:

The default method "CSDP" is an interior point method:

"DSDP" is an alternative interior point method:

"SCS" uses a splitting conic solver method:

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

Compute exact and approximate solutions:

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

"SCS" has a default tolerance of :

When the default method "CSDP" produces a message, try "DSDP" first:

In this case, "DSDP" succeeds in finding a good solution:

"SCS" with a default tolerance of 10^-3 is an alternative method to try:

The quality of the result with "SCS" can often be improved with a smaller Tolerance:

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 result, but may take longer to compute:

A smaller tolerance takes longer:

The tighter tolerance gives a more precise answer:

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

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

Find that minimizes the largest eigenvalue of a symmetric matrix that depends linearly on the decision variables , . The problem can be formulated as 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 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 linear in symmetric matrix :

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 linear in matrix :

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:

Minimize . Using an auxiliary variable , transform the problem to minimize such that . This is the same as :

A Schur complement condition says that if , a block matrix iff . Thus ⧦ for and for , since then must be 0:

Use the constraint directly and it will automatically convert into semidefinite form:

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

Check that implies :

Using the Schur complement condition, iff . Use Inactive[Plus] for constructing the constraints to avoid threading:

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 :

Since λ≥0, :

Minimize subject to . Convert the objective into a linear function with the additional constraint , which is equivalent to :

Minimize subject to . Convert the objective into a linear function using and the additional constraints :

Minimize subject to . Convert the objective into a linear function with the additional constraint , which is equivalent to :

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

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

Find the coefficients of a fifth-order polynomial by minimizing that fits a discrete dataset:

Select the polynomial bases and construct the input matrix using DesignMatrix and output vector:

Using an auxiliary variable , the objective is transformed to minimize such that , which is equivalent to as shown under Basic Modeling Transformations:

Compare fit with data:

Find an approximating function to discrete data that varies on a logarithmic scale by minimizing using Chebyshev bases:

Select Chebyshev basis functions and compute their values at the random data points:

Since the data is on a logarithmic scale, direct data-fitting is not ideal. Instead, transform the problem to minimize . Using auxiliary variable , minimize such that . This constraint is equivalent to :

Find the coefficients of the approximating function:

The resulting fit is:

Visualize the fit:

The data-fitting can also be obtained directly using the function Fit. However, without the log transformation, there are significant oscillations in the approximating function:

Represent a given bivariate polynomial in terms of 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:

Cardinality constrained least squares: minimize such that has at most nonzero elements:

Let be a decision vector such that if , then is nonzero. The decision constraints are:

To model constraint when , chose a large constant such that :

Using an auxiliary variable , the objective is transformed to minimize such that , which is equivalent to :

Solve the cardinality constrained least-squares problem:

The subset selection can also be done more efficiently with Fit using regularization. First, find the range of regularization parameters that uses at most basis functions:

Find the nonzero terms in the regularized fit:

Find the fit with just these basis terms:

Find the best subset of functions from a candidate set of functions to approximate given data:

The approximating function will be :

A maximum of 5 basis functions are to be used in the final approximation:

The coefficients associated with functions that are not chosen must be zero:

Find the best subset of functions:

Compare the resulting approximating with the given data:

Find the smallest disk centered at of radius that encloses a set of points:

For each point , the constraint must be satisfied. This constraint is equivalent to . Use Inactive when forming the constraints:

Find the enclosing disk by minimizing the radius :

Visualize the enclosing region:

The minimal area bounding disk can also be found using BoundingRegion:

Find the smallest ellipse parametrized as that encompasses a set of points by minimizing the area:

For each point , the constraint must be satisfied:

The area is proportional to . Applying the monotone function Log, the function to minimize is . This in turn is equivalent to minimizing :

Convert the parameterized ellipse into the explicit form :

A bounding ellipse, not necessarily minimal area, can be found using BoundingRegion:

The optimal ellipse has a smaller area:

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 parameterized ellipse into the explicit form :

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

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 :

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

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

Convert the parameterized ellipse into the explicit form as :

Find the center and radius of a disk given by that encloses three ellipses of the form :

Using S-procedure, it can be shown that the disk contains the ellipses iff :

The goal is to minimize the radius given by . Using auxiliary variable , the objective is to minimize such that , which can be written as :

Find the center and radius of the disk:

The disk is given by:

Convert the quadratic form of the ellipse to the explicit form :

Visualize the result:

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

Using 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 an ellipse that separates two groups of points and :

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

Find the coefficients of the separating ellipsoid:

Visualize the result:

Find an ellipse that is as close as possible to a circle that separates two groups of points and :

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

For the ellipsoid to be as close as possible to a circle, the constraint :

Find the coefficients of the separating ellipsoid by minimizing :

Visualize the result:

The Lovász number, computable using semidefinite optimization, is used as a bound for hard compute graph invariants:

The Lovász number is an upper bound for the Shannon capacity of a graph:

According to the Lovász sandwich theorem:

The Lovász number for a graph is given by , where , and for . It can be written in a dual semidefinite form: , subject to and and 0 elsewhere:

Compare to the exact Lovász number values from GraphData:

Find the approximate result for when the exact result is not available:

Find for a random graph:

The max-cut problem determines a subset of the vertices of a graph, for which the sum of the weights of the edges that cross from to its complement is maximized. Let for and for . Maximize , where and is the Laplacian matrix of the graph:

For smaller cases, the max-cut problem can be solved exactly, but this is impractical for larger graphs since in general the problem has NP-complete complexity:

The problem minimizes , where is a symmetric rank-1 positive semidefinite matrix, with for each , equivalent to , where is the matrix with at the diagonal position and 0 everywhere else. To make the solution practical, solve a relaxed problem where the rank-1 condition is eliminated. For such , a cut is constructed by randomized rounding: decompose , let be a uniformly distributed random vector of the unit norm and let . For demonstration, a function is defined that shows the relaxed value, the rounded value and the graph with the vertices in shown as red:

Find an approximate max cut using the previously shown procedure, and compare with the exact result:

Find the max cut for a grid graph:

Find the max cut for a random graph:

Compare timings for the relaxed and exact algorithms for a Peterson graph:

Find a subset of specified graph with vertices such that the sum of the weights of the edges that cross from to its complement is maximized. Specify the graph:

The objective is to maximize , where is a symmetric rank-1 positive semidefinite matrix and is the Laplacian matrix of the graph:

Let for and for ; then and :

Drop the rank-1 matrix assumption and solve the resulting max-cut problem:

Extract the subsets and :

Display the subsets on the graph:

Show that a linear dynamical system will be asymptotically stable for any initial condition. The system is said to be stable iff there exists a positive definite matrix such that where is called the Lyapunov function:

Differentiating the Lyapunov function gives . Therefore, the stability conditions are :

Find a matrix :

The eigenvalues of are all negative, making the matrix negative definite, which proves stability:

Since the analytic solution to the system is , numerically verify that any system will go to zero for any initial condition. Take for this simulation:

Find a controller , such that the closed-loop system is stable:

Using Lyapunov stability theorem, the objective is to find a matrices such that the stability constraints are satisfied. Letting , the first constraint becomes a proper semidefinite constraint :

Find the matrices :

The control matrix can be computed as :

The closed-loop system is stable if the real parts of the eigenvalues of are negative:

Perform a numerical run to see that the system is stable:

Find a Lyapunov function of the dynamical system :

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

Construct the matrix :

For stability, :

Match the coefficients such that they are all positive for and negative for :

Find the matrix :

The Lyapunov function is given by:

Visualize the Lyapunov function. The minimal location of the function matches with the location of the attractor:

Design a minimum-weight truss that is anchored on one end of the wall and must withstand a load on the other end:

The truss can be modeled using links and nodes. Each node is connected to a neighboring node by a link. Specify the node positions :

Specify the nodes that are anchored to the wall:

Specify the node at which the load is applied:

Specify the nodes that are connected to each other through a link and compute the length of each link:

Visualize the unoptimized truss:

The links are circular bars. Each link must be formed from one out of a group of bars of cross sections . Let be a decision vector for each link , such that if , then bar is selected. For link , the area is then defined as . The objective is to minimize the weight:

The bar selection constraint is:

Only one bar must be selected for each link. The binary constraint is:

Find the indices of the nodes that are not anchored:

The stiffness matrix of the system is given by , where is the total number of nodes, is the number of nodes that are anchored and is the set of all the links. The vector if ; if , else 0:

Let be the force vector for the entire system. At each of the nodes that is not anchored, the force is . The node where force is applied is :

Let be the maximum allowable deflection at any node. Let be displacement of node ; then is satisfied if , where is the stiffness matrix associated with link :

Collect all the variables:

Find the optimal structure of the truss:

Extract the links that are part of the optimal truss:

Visualize the optimal truss. The links that are part of the optimal truss are colored-coded based on the rod area being used: