Minimize subject to the constraints :

Get the minimizing value using solution property "PrimalMinimumValue":

Minimize subject to the constraint :

Get the minimizing value and minimizing vector using solution property:

Minimize subject to the constraint :

Define the objective as and constraints as :

Solve using matrix-vector inputs:

Minimize subject to the equality constraint and the inequality constraint :

Define objective as and constraints as and :

Solve using matrix-vector inputs:

Minimize subject to the constraints :

Define the objective as and constraints as :

Solve using matrix-vector inputs:

Minimize subject to the constraints :

Specify constraints using VectorGreaterEqual () and VectorLessEqual ():

Minimize subject to . Use a vector variable and constant parameter equations:

Minimize subject to . Use NonNegativeReals to specify the constraint :

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 constraints using NonPositiveIntegers ():

Specify complex variables using Complexes:

Use a Hermitian matrix in the objective with real-valued variables:

Use a Hermitian matrix in the objective and complex variables:

Minimize subject to the constraint :

Get vector output using "PrimalMinimizer":

Get the rule-based result using "PrimalMinimizerRules":

Get the minimizing value of the optimization using "PrimalMinimumValue":

Obtain the primal minimum value using symbolic inputs:

Extract the matrix-vector inputs of the optimization problem:

Solve using the matrix-vector form:

Recover the symbolic primal value by adding the objective constant:

Find the slack for inequalities and equalities associated with a minimization problem:

Get the minimum value and the constraint matrices:

The slack for inequality constraints is a vector such that :

The slack for the equality constraints is a vector such that :

The equality slack is typically a zero vector:

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:

So it has 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 and constraint matrices and vectors:

The dual problem is to maximize subject to :

Get the dual maximum value directly using solution properties:

Get the dual maximizer directly using solution properties:

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

The first vector is inequality sensitivity and the second is equality sensitivity:

Consider new constraints where is the relaxation:

The approximate new optimal value is given by:

Compare to directly solving the relaxed problem:

Each sensitivity is associated with an inequality or equality constraint:

Extract the constraints:

The inequality constraints and their associated sensitivity:

The equality constraints and their associated sensitivity:

The change in optimal value due to constraint relaxation is proportional to the sensitivity value:

Compute the minimal value and constraint sensitivity:

A zero sensitivity will not change the optimal value if the constraint is relaxed:

A negative sensitivity will decrease the optimal value:

A positive sensitivity will increase the optimal value:

The "ConstraintSensitivity" is related to the dual maximizer of the problem:

The inequality sensitivity satisfies , where is the dual inequality maximizer:

The equality sensitivity satisfies , where is the dual equality maximizer:

The method "COIN" uses the COIN library:

"CSDP" and "DSDP" reduce to semidefinite optimization:

"SCS" reduces to conic optimization:

"PolyhedralApproximation" approximates the objective using linear constraints:

For least-squares-type quadratic problems, "CSDP" and "DSDP" will be slower than "COIN" or "SCS":

Solve the least-squares problem using method "COIN":

Solve the problem using method "CSDP":

For least-squares-type quadratic problems, "CSDP","DSDP" and "PolyhedralApproximation" will be slower than "COIN" or "SCS":

Solve the least-squares problem using method "COIN":

Solve the problem using method "CSDP":

Solve the problem using method "PolyhedralApproximation":

Different methods may give different results for problems with more than one optimal solution:

Minimizing a concave function can be done only using Method"COIN":

Other methods cannot be used because they require the factorization of the objective matrix:

Get more accurate results at the cost of higher computation time with "Quality" setting:

Use "Speed" to get results quicker but at the cost of quality:

A smaller Tolerance setting gives a more precise result:

Find the error between computed and exact minimum value using different Tolerance settings:

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

A smaller Tolerance setting gives a more precise answer, but typically takes longer to compute:

A smaller tolerance takes longer:

The tighter tolerance gives a more precise answer:

MachinePrecision is the default for the WorkingPrecision option in QuadraticOptimization:

With WorkingPrecisionAutomatic, QuadraticOptimization infers the precision to use from input:

QuadraticOptimization can compute results using arbitrary-precision numbers:

If the specified precision is less than the precision of the input arguments, a message is issued:

If a high-precision result cannot be computed, a message is issued and a MachinePrecision result is returned:

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

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

Minimize by converting the objective function into :

Construct the objective function by expanding :

Since QuadraticOptimization minimizes , the matrix is multiplied by 2:

The minimal value of the original function is recovered as :

QuadraticOptimization directly performs this transformation. Construct the objective function using Inactive to avoid threading:

Minimize subject to the constraints . Transform the objective function to and solve the problem:

Recover the original minimum value using transformation :

Minimize subject to the constraints . The constraint can be transformed to :

Minimize subject to the constraints . The constraint can be interpreted as . Solve the problem with each constraint:

The optimal solution is the minimum of the two solutions:

Minimize subject to , where is a non-decreasing 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 the function to the primal minimum value:

Minimize , subject to :

Since , the solution is the true solution only if the primal minimum value is greater than 0. The true minimum value can be obtained by applying the function to the primal minimum value:

Find a linear fit to discrete data by minimizing :

Construct the factored quadratic matrix using DesignMatrix:

Find the coefficients of the line:

Compare fit with data:

Find a quadratic fit to discrete data by minimizing :

Construct the factored quadratic matrix using DesignMatrix:

Find the coefficients of the quadratic curve:

Compare fit with data:

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

Construct the factored quadratic matrix using DesignMatrix:

The two equality constraints are:

Find the coefficients of the line:

Compare fit with data:

Find an interpolating function to noisy data using bases :

The interpolating function will be :

Find the coefficients of the interpolating function:

Compare the fit to the data:

Minimize subject to the constraints :

Compare with the unconstrained minimum of :

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 :

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 is 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 a plane that separates two groups of 3D points and :

For separation, set 1 must satisfy and set 2 must satisfy . Find the hyperplane by minimizing :

The distance between the planes and is :

The plane separating the two groups of points is:

Plot the plane separating the two datasets:

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 surface by minimizing :

The polynomial separating the two groups of points is:

Plot the polynomial separating the two datasets:

Find a point closest to the point that lies on the planes and :

Find the point closest to by minimizing . Use Inactive Plus when constructing the objective:

Find the distance between two convex polyhedra:

Show the nearest points with the line connecting them:

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

The original minimization problem is to minimize subject to . The dual of this problem is to maximize subject to :

Solve the dual maximization problem:

The center of the minimal enclosing ball is :

The radius of the minimal enclosing ball is Sqrt of the maximum value:

Visualize the enclosing ball:

The minimal enclosing ball can be found efficiently using BoundingRegion:

Find the number of stocks to buy from four stocks, such that a minimum $1000 dividend is received and risk is minimized. The expected return value and the covariance matrix associated with the stocks are:

The unit price for the four stocks is $1. Each stock can be allocated a maximum of $2500:

The investment must yield a minimum of $1000:

A negative stock cannot be bought:

The total amount to spend on each stock is found by minimizing the risk given by :

The total investment to get a minimum of $1000 is:

Find the number of stocks to buy from four stocks, with an option to short-sell such that a minimum dividend of $1000 is received and the overall risk is minimized:

The capital constraint and return on investment constraints are:

A short-sale option allows the stock to be sold. The optimal number of stocks to buy/short-sell is found by minimizing the risk given by the objective :

The second stock can be short-sold. The total investment to get a minimum of $1000 due to the short-selling is:

Without short-selling, the initial investment will be significantly greater:

Find the best combination of six stocks to invest in out of a possible 20 candidate stocks, so as to maximize return while minimizing risk:

Let be the percentage of total investment made in stock . The return is given by , where is a vector of the expected return value of each individual stock:

Let be a decision vector such that if , then that stock is bought. Six stocks have to be chosen:

The percentage of investment must be greater than 0 and must add to 1:

Find the optimal combination of stocks that minimizes the risk given by and maximizes return:

The optimal combination of stocks is:

The percentages of investment to put into the respective stocks are:

Find the distribution of capital to invest in six stocks to maximize return while minimizing risk:

Let be the percentage of total investment made in stock . The return is given by , where is a vector of expected return value of each individual stock:

The risk is given by , and is a risk-aversion parameter:

The objective is to maximize return while minimizing risk for a specified risk-aversion parameter :

The percentage of investment must be greater than 0 and 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 percentage of investment for a specified number of risk-aversion parameters:

Increasing the risk-aversion parameter leads to stock diversification to reduce the risk:

Increasing the risk-aversion parameter leads to a reduced expected return on investment: