Minimize subject to with parameter :

Minimize subject to and parameter :

Minimize subject to and parameters :

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 VectorGreaterEqual to express several GreaterEqual inequality constraints at once:

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

Specify the constraints using a combination of scalar and vector inequalities:

An equivalent form using scalar inequalities:

Use vector variables and vector inequalities to specify the problem:

Minimize the function subject to the constraint and parameters :

Use Indexed to access components of a vector variable, e.g. :

Use constant parameter equations to specify the coefficients of the objective and constraints:

Minimize the function subject to and parameter :

Use Vectors[n,Reals] to specify the dimensions of vector variables when needed:

Specify non-negative constraints using NonNegativeReals ():

An equivalent form using vector inequalities:

Specify non-positive constraints using NonPositiveReals ():

Specify bounds using Interval:

Use Vectors[n,Reals] to specify vector parameters:

Specify constraints on the parameter to ensure feasibility:

The constraint cannot be satisfied with :

Specify constraints on the parameter to ensure that the objective and/or constraints are convex:

Without the additional constraint on the parameter, the convexity cannot be determined:

Specify integer domain constraints using Integers:

Visualize the dependence of parameter on the minimum value:

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 integer domain on parameters using Integers:

A message is issued if noninteger values are given for the parameters:

Specify integer domain on vector variable parameters using Vectors[n,Integers]:

Specify complex variables and parameters using Complexes:

Minimize a real objective with complex variables and complex constraints :

Let . Expanding out the constraints into real components gives:

Specify a real-valued objective with a vector parameter :

Solve the problem with :

Specify complex -dimensional vector variables and parameters using Vectors[n,Complexes]:

Plot the derivative of the minimum value with respect to a parameter:

Find the sensitivity of least-squares fit coefficients to a data outlier:

Define a function to give the fit coefficients' residual, slope and intercept {r,m,b} as a function of the measured values :

Show the data with the least-squares fit:

Define a function that gives the derivative with respect to the first measured value and compute it for the data values:

Define a function that gives the derivative with respect to the fifth measured value and compute it for the data values:

The fit is much less sensitive to the fifth than the first value.

Show how a minimizer changes with the direction of the objective vector :

Define a function to get the Jacobian matrix of the minimizer with respect to :

Consider a parametrization of the direction . The derivative with respect to is given by :

Show the minimizer and its derivative vector with respect to with the convex region defined by the constraints for a few values of :

The derivative vector is tangent to the boundary.

Minimize Cos(3α)x+Sin(2α)y over a disk for varying parameter :

Get the primal minimizer as a rule:

Get the primal minimizer as a vector:

Show how the position of the minimizer changes with :

Get the primal minimal value:

Plot the minimal value as a function of :

Use the epigraph transformation to get the primal minimum value as part of the primal minimizer:

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

Extract the minimizer and conic constraint affine lists:

Verify that the slack satisfies with :

Minimize subject to and and parameters :

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:

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

Show the distance as a function of :

Find the minimum distance:

Show the ellipse found with the closest point:

Visualize the positions of the two points with respect to the disk:

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

The area is given by the negative of the function:

Plot the maximum area as a function of :

Maximize the maximum area:

The best shape is, not surprisingly, a square. Note the message is because the values of the function are approximated and so have significant variation near the maximum:

Find the minimum distance between two disks of radius 1 centered at and , where is a vector parameter. Let be a point on disk 1. Let be a point on disk 2. The objective is to minimize subject to constraints :

For example, if :

Now plot the minimum distance between the two disks for arbitrary parameter values in :

Find an approximate distribution for linear fit coefficients when sample error is normally distributed:

The ParametricFunction gives the fit coefficients for a particular error instance:

Generate 1000 instances of the samples:

Compute fit coefficients with no regularization for each of the instances and find the distribution of the coefficients:

Show the PDF of the distributions, along with a histogram of the data:

Compare with an L1 fit:

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 minimum residual:

Get the fit coefficients corresponding to the minimum residual:

Show the fit with the data:

The optimal width basis elements have noticeable overlap:

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:

Construct a parametric function:

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:

Minimize subject to . The minimizing function integral can be approximated using the trapezoidal rule. The discretized objective function will be :

The constraint can be discretized using finite differences:

The constraints can be represented using the Indexed function:

The constraints can be discretized using finite differences, and only the first and last rows are used:

Solve the discretized trajectory problem:

Visualize the result for a range of parameters :

Visualize the second derivative for a range of parameters :

Find the shortest path between two points while avoiding obstacles. Specify the obstacle:

To avoid potential crossings at the edges, the region can be inflated, but the result will be valid for the original region:

Obtain a set of linear inequalities that enclose the inflated obstacle:

The starting point is given by the parameter , and the ending point is fixed:

The path can be discretized using points. Let represent the position vector:

The objective is to minimize . Let . The objective is transformed to :

Specify the starting and ending point constraints:

The distance between any two subsequent points should not be too large:

A point is outside the object if at least one element of is less than zero. To enforce this constraint, let be a decision vector and be the element of such that ; then and is large enough such that :

Generate a parametric function for solving the optimization problem for different starting points:

Compute and display the path for various starting points:

Make a function test inclusion of points in a set defined by constraints:

Set up a function to test inclusion in the set defined by :

The origin is included:

Test a set of randomly generated points: