Use NMaximize with method "MOSEK" to maximize 1-TemplateBox[{{x, +, {2, y}}}, Abs] subject to linear constraints:

Use ConvexOptimization to minimize over a disk centered at with radius

Get the minimum value and the minimizing vector using solution properties:

Use ConicOptimization to minimize subject to TemplateBox[{{x, +, , y}}, Abs]^(1.5)<=t and {x,y} in Disk[{1,1}]:

Get the dual maximizer:

Use SemidefiniteOptimization to minimize subject to the positive semidefinite matrix constraint (x 1; 1 y)_(TemplateBox[{2}, SemidefiniteConeList])0:

Find the solution:

Use SecondOrderConeOptimization to minimize subject to :

Define the objective as and the constraints as TemplateBox[{{{{a, _, i}, ., x}, +, {b, _, i}}}, Norm]<=alpha_i.x+beta_i,i=1,2:

Specify the equality constraint as:

Solve using matrix-vector inputs:

Use QuadraticOptimization to minimize minimize subject to and :

Define objective as and constraints as and :

Solve using matrix-vector inputs:

Use LinearOptimization to minimize subject to :

Combine the coefficients into and use a vector variable :

Use GeometricOptimization to maximize the area of a rectangle such that the perimeter is at most 1:

Minimize Total [x] subject to the constraint using vector variable with non-negative values:

Minimize Total [x] subject to the constraint with a non-negative integer-valued vector:

Minimize Total [x] subject to the constraint using a vector variable :

Minimize the sum of the integer-valued coordinates of a point lying within a 1000-dimensional unit ball:

Minimize for a sparse symmetric semidefinite matrix , subject to constraint :

Minimize subject to the constraint for large sparse matrices , and :

Minimize x.Q.x+Total [x] for a sparse symmetric semidefinite matrix , subject to Total [x]≥1:

Given an matrix with non-negative real entries, find a diagonal matrix with positive entries that minimizes the sum of squares (the Frobenius norm squared) of the similar matrix :

Let be the diagonal entries of . Since is positive, the entries of are , so the entries of the product are :

Find that minimizes :