### Running the CVX Quick Start Script

Source: https://cvxr.com/cvx/doc/quickstart.html

Navigate to the CVX examples directory and run the `quickstart` script to follow along with the examples.

```matlab
cd D:\Matlab\cvx\examples
quickstart

```

--------------------------------

### Install CVX Professional License

Source: https://cvxr.com/cvx/doc/_sources/install.txt

When installing CVX, include the path to your license file as an argument to `cvx_setup` to install a CVX Professional license. This example is for Mac.

```matlab
cd ~/MATLAB/cvx
cvx_setup ~/licenses/cvx_license.mat
```

--------------------------------

### Run cvx_setup on Windows

Source: https://cvxr.com/cvx/doc/_sources/install.txt

Navigate to the CVX directory and run the setup command in MATLAB. This is for Windows users.

```matlab
cd C:\personal\cvx
cvx_setup
```

--------------------------------

### Run CVX Setup Script

Source: https://cvxr.com/cvx/doc/_sources/funcref.txt

The setup script used to install and configure CVX.

```matlab
cvx_setup
```

--------------------------------

### Install CVX with Professional License

Source: https://cvxr.com/cvx/doc/install.html

Install CVX with a professional license by providing the license file path to the setup command.

```matlab
cd ~/MATLAB/cvx
cvx_setup ~/licenses/cvx_license.mat

```

--------------------------------

### Run cvx_setup on Linux/Mac

Source: https://cvxr.com/cvx/doc/_sources/install.txt

Navigate to the CVX directory and run the setup command in MATLAB. This is for Linux or Mac users.

```matlab
cd ~/MATLAB/cvx
cvx_setup
```

--------------------------------

### Get CVX Installation Directory

Source: https://cvxr.com/cvx/doc/_sources/funcref.txt

Returns the directory where CVX is installed.

```matlab
cvx_where
```

--------------------------------

### Begin CVX Model

Source: https://cvxr.com/cvx/doc/_sources/funcref.txt

Starts a new CVX model. If a model is already in progress, it will issue a warning and clear it.

```matlab
cvx_begin
```

--------------------------------

### Begin CVX Model Quietly

Source: https://cvxr.com/cvx/doc/_sources/solver.txt

Start a CVX model while suppressing screen output from the solver. This is useful for integrating CVX into larger algorithms.

```matlab
cvx_begin quiet
    ...
cvx_end
```

--------------------------------

### Objective with log_det and trace

Source: https://cvxr.com/cvx/doc/advanced.html

This example shows an objective function that cannot be simplified by removing the logarithm from log_det due to the presence of other terms like trace(C*X).

```matlab
minimize( log_det(X) + trace(C*X) )

```

--------------------------------

### Minimize L1 Norm using linprog (MATLAB)

Source: https://cvxr.com/cvx/doc/_sources/quickstart.txt

This snippet shows the manual setup for minimizing the L1 norm using MATLAB's linprog function. It defines the objective function, equality constraints, and bounds.

```matlab
f    = [ zeros(n,1); ones(m,1);  ones(m,1)  ];
Aeq  = [ A,          -eye(m),    +eye(m)    ];
lb   = [ -Inf(n,1);  zeros(m,1); zeros(m,1) ];
xzz  = linprog(f,[],[],Aeq,b,lb,[]);
x_l1 = xzz(1:n,:) - xzz(n+1:end,:);
```

--------------------------------

### Switch Between Multiple Gurobi Versions

Source: https://cvxr.com/cvx/doc/_sources/gurobi.txt

If multiple Gurobi versions are installed, CVX appends a numeral to the solver name, allowing you to switch between them. Use these commands to select specific versions.

```matlab
cvx_solver gurobi
cvx_solver gurobi_2
cvx_solver gurobi_3
```

--------------------------------

### Implementing a Convex Function via DCP Rules

Source: https://cvxr.com/cvx/doc/_sources/advanced.txt

This example shows how to define a new convex function, 'deadzone', within CVX by composing existing DCP-compliant functions like 'abs' and 'max'. The function works both inside and outside CVX if its operations adhere to DCP rules.

```matlab
function y = deadzone( x )
y = max( abs( x ) - 1, 0 )
```

--------------------------------

### Declare and Use Indexed Dual Variables

Source: https://cvxr.com/cvx/doc/_sources/advanced.txt

Model a semidefinite program using indexed dual variables to handle a variable number of constraints. This example uses a for loop to assign dual variables.

```matlab
cvx_begin
    variable X( n, n ) symmetric
    minimize( ( n - 1 : -1 : 0 ) * diag( X ) );
    for k = 0 : n-1,
        sum( diag( X, k ) ) == b( k+1 );
    end
    X == semidefinite(n);
cvx_end
```

--------------------------------

### CVX Constraint: Valid Convex Form

Source: https://cvxr.com/cvx/doc/_sources/support.txt

This example shows a mathematically equivalent form of a constraint that is accepted by CVX because it adheres to the Disciplined Convex Programming (DCP) ruleset. It demonstrates how to correctly express convex constraints.

```matlab
sqrt( sum( square( x ) ) ) <= 1
```

--------------------------------

### Set Gurobi as Default Solver

Source: https://cvxr.com/cvx/doc/_sources/gurobi.txt

Change the active solver to Gurobi for the current session and save this preference for future MATLAB sessions. This ensures Gurobi is automatically selected when CVX starts.

```matlab
cvx_solver gurobi
cvx_save_prefs
```

--------------------------------

### Get Current Precision Levels

Source: https://cvxr.com/cvx/doc/solver.html

Call `cvx_precision` with no arguments to print the current tolerance levels to the screen, or as a function to return them in a 3-element row vector.

```matlab
cvx_precision
```

```matlab
tol = cvx_precision;
```

--------------------------------

### List Custom Settings for All Solvers

Source: https://cvxr.com/cvx/doc/solver.html

This command displays all custom settings that have been provided for every solver supported by CVX. It is useful for a comprehensive overview of all potential customizations.

```matlab
cvx_solver_settings -all

```

--------------------------------

### Clear CVX Model

Source: https://cvxr.com/cvx/doc/_sources/funcref.txt

Clears any model being constructed. Useful when an error has been made and it is necessary to start from the beginning.

```matlab
cvx_clear
```

--------------------------------

### List Custom Settings for Active Solver

Source: https://cvxr.com/cvx/doc/solver.html

Use this command to view the custom settings currently applied to the active solver. This helps in understanding the current configuration before making changes.

```matlab
cvx_solver_settings

```

--------------------------------

### Basic Semidefinite Program in CVX

Source: https://cvxr.com/cvx/doc/advanced.html

This snippet demonstrates a basic semidefinite program in CVX without indexed dual variables. It sets up the optimization variable, objective function, and constraints.

```matlab
cvx_begin
    variable X( n, n ) symmetric
    minimize( ( n - 1 : -1 : 0 ) * diag( X ) );
    for k = 0 : n-1,
        sum( diag( X, k ) ) == b( k+1 );
    end
    X == semidefinite(n);
cvx_end

```

--------------------------------

### Maximize Norm Error in CVX

Source: https://cvxr.com/cvx/doc/quickstart.html

Example of an invalid CVX specification attempting to maximize a convex function, resulting in an error.

```matlab
maximize( norm(A*x-b) );

```

--------------------------------

### Convexity of Square Root of Sum of Squares

Source: https://cvxr.com/cvx/doc/_sources/dcp.txt

An example of an expression that is convex but may not satisfy the DCP rules directly, illustrating the limitations of the ruleset.

```Python
sqrt( sum( square( x ) ) )
```

--------------------------------

### List Custom Solver Settings

Source: https://cvxr.com/cvx/doc/_sources/solver.txt

Displays custom settings for the active CVX solver or all solvers.

```matlab
cvx_solver_settings
```

```matlab
cvx_solver_settings -all
```

--------------------------------

### Retrieve Gurobi License Key

Source: https://cvxr.com/cvx/doc/_sources/gurobi.txt

Use this command to retrieve your Gurobi license key. This is a wrapper around Gurobi's own grbgetkey script. Ensure you are connected to your university network if you are an academic user.

```matlab
cvx_grbgetkey xxxxxxxx-xxxx-xxxx-xxxx-xxxxxxxxxxxx
```

--------------------------------

### Get current CVXPY precision levels

Source: https://cvxr.com/cvx/doc/_sources/solver.txt

Call `cvx_precision` without arguments to print the current tolerance levels to the screen. This is useful for inspecting the active precision settings.

```matlab
cvx_precision
```

--------------------------------

### Using Assignment for Intermediate Calculations

Source: https://cvxr.com/cvx/doc/basics.html

Demonstrates using assignment to store an intermediate affine expression 'z' which is then used in multiple constraints.

```matlab
variables x y
z = 2 * x - y;
square( z ) <= 3;
quad_over_lin( x, z ) <= 1;

```

--------------------------------

### Maximize Volume Box using GP Mode

Source: https://cvxr.com/cvx/doc/_sources/gp.txt

This snippet demonstrates how to define a geometric program to find the maximum volume of a box subject to area and ratio constraints using CVX's GP mode. It requires `cvx_begin gp` to activate the mode.

```matlab
cvx_begin gp
    variables w h d
    maximize( w * h * d )
    subject to
        2*(h*w+h*d) <= Awall;
        w*d <= Afloor;
        alpha <= h/w >= beta;
        gamma <= d/w <= delta;
cvx_end
```

--------------------------------

### CVX Constraint Violation Example

Source: https://cvxr.com/cvx/doc/support.html

This code demonstrates a common scenario where a constraint is rejected by CVX due to violating the Disciplined Convex Programming (DCP) ruleset. The error indicates an invalid comparison between a convex expression and a real constant.

```matlab
variable x(10)
norm(x) == 1

```

--------------------------------

### Retrieve Gurobi License Key

Source: https://cvxr.com/cvx/doc/gurobi.html

Use this command to retrieve your Gurobi license key. Ensure you replace the placeholder with your actual 32-digit Gurobi key. This step is crucial for enabling Gurobi within CVX.

```bash
cvx_grbgetkey xxxxxxxx-xxxx-xxxx-xxxx-xxxxxxxxxxxx

```

--------------------------------

### Using Assignments for Intermediate Calculations

Source: https://cvxr.com/cvx/doc/_sources/basics.txt

Demonstrates how to use assignments to store intermediate affine expressions that are then used in multiple constraints. This is useful for breaking down complex expressions.

```cvxpy
variables x y
z = 2 * x - y;
square( z ) <= 3;
quad_over_lin( x, z ) <= 1;
```

--------------------------------

### Chebyshev Norm Minimization with MATLAB linprog

Source: https://cvxr.com/cvx/doc/_sources/quickstart.txt

Solves the Chebyshev (infinity) norm minimization problem using MATLAB's linprog. This requires reformulating the problem into a linear program.

```matlab
f    = [ zeros(n,1); 1          ];
Ane  = [ +A,         -ones(m,1)  ; ...
         -A,         -ones(m,1) ];
bne  = [ +b;         -b         ];
xt   = linprog(f,Ane,bne);
x_cheb = xt(1:n,:);
```

--------------------------------

### Display CVX Version Information

Source: https://cvxr.com/cvx/doc/_sources/funcref.txt

Prints information about the current versions of CVX, Matlab, and the operating system. When submitting bug reports, please include the output of this command.

```matlab
cvx_version
```

--------------------------------

### Convexity of Affine Composition

Source: https://cvxr.com/cvx/doc/_sources/dcp.txt

Demonstrates how an affine transformation of a convex function results in a convex function. This is a fundamental rule in DCP.

```Python
square( a' * x + b )
```

--------------------------------

### Commercial Solver Directories

Source: https://cvxr.com/cvx/doc/license.html

Files within these directories are licensed by third-party providers for inclusion in the CVX package and are subject to the CVX Professional License.

```text
gurobi/
mosek/
```

--------------------------------

### Minimize L2 Norm with Equality and L-infinity Constraints using CVX

Source: https://cvxr.com/cvx/doc/_sources/quickstart.txt

This snippet demonstrates minimizing the L2 norm while satisfying an equality constraint and an L-infinity norm constraint. CVX converts this into an SOCP.

```matlab
cvx_begin
    variable x(n);
    minimize( norm(A*x-b) );
    subject to
        C*x == d;
        norm(x,Inf) <= 1;
cvx_end
```

--------------------------------

### Constrain Matrix with Semidefinite Cone using '<In>'

Source: https://cvxr.com/cvx/doc/basics.html

An alternative syntax using the '<In>' pseudo-operator to constrain a matrix to the semidefinite cone. This is equivalent to using the equality constraint.

```matlab
X <In> semidefinite(n);
```

--------------------------------

### Create a New Custom Solver Setting

Source: https://cvxr.com/cvx/doc/solver.html

This syntax allows you to define a new custom setting for the current solver. The `{name}` must be a valid MATLAB variable name, and `{value}` can be any valid MATLAB object. CVX does not validate the value itself.

```matlab
cvx_solver_settings( '{name}', {value} )

```

--------------------------------

### BibTeX Citation for Graph Implementations Paper

Source: https://cvxr.com/cvx/doc/_sources/citing.txt

This BibTeX entry is for citing the paper on graph implementations for nonsmooth convex programs by M. Grant and S. Boyd. It includes details about the book, series, editors, publisher, pages, year, and a URL.

```bibtex
@incollection{gb08,
  author    = {M. Grant and S. Boyd},
  title     = {Graph implementations for nonsmooth convex programs},
  booktitle = {Recent Advances in Learning and Control},
  series    = {Lecture Notes in Control and Information Sciences},
  editor    = {V. Blondel and S. Boyd and H. Kimura},
  publisher = {Springer-Verlag Limited},
  pages     = {95--110},
  year      = 2008,
  note      = {\url{http://stanford.edu/~boyd/graph_dcp.html}}
}
```

--------------------------------

### Formulating a Positive Semidefinite Constraint

Source: https://cvxr.com/cvx/doc/_sources/advanced.txt

This snippet demonstrates how to express a problem of making a matrix positive semidefinite by adding a diagonal matrix. It highlights a common pitfall with unsymmetric matrices leading to 'Overdetermined' status.

```matlab
n = size(W,1);
cvx_begin
    variable D(n,n) diagonal;
    minimize( trace( D ) );
    subject to
        W + D == semidefinite(n);
cvx_end
```

--------------------------------

### Sum of Convex and Concave Functions

Source: https://cvxr.com/cvx/doc/dcp.html

Illustrates the sum of a convex function (square) and a concave function (sqrt). CVX classifies these based on the properties of the sum.

```python
sum( square( x ) )
```

```python
sum( sqrt( x ) )
```

--------------------------------

### Minimize L1 Norm using CVX

Source: https://cvxr.com/cvx/doc/_sources/quickstart.txt

This snippet demonstrates how to minimize the L1 norm using CVX. CVX automatically transforms this into an LP.

```matlab
cvx_begin
    variable x(n)
    minimize( norm(A*x-b,1) )
cvx_end
```

--------------------------------

### Set MOSEK as Default Solver

Source: https://cvxr.com/cvx/doc/_sources/mosek.txt

Change the active solver to MOSEK for the current session and save this preference for future MATLAB sessions.

```matlab
cvx_solver mosek
cvx_save_prefs
```

--------------------------------

### SDP Mode LMI Constraint in CVX

Source: https://cvxr.com/cvx/doc/sdp.html

This snippet demonstrates how the same Hermitian semidefinite constraint can be expressed more concisely using CVX's SDP mode with the '>=' operator.

```matlab
cvx_begin sdp
    variable Z(n,n) hermitian toeplitz
    dual variable Q
    minimize( norm( Z - P, 'fro' ) )
    Z >= 0 : Q;
cvx_end

```

--------------------------------

### Product of Inverses

Source: https://cvxr.com/cvx/doc/_sources/funcref.txt

Calculates the product of the inverses of elements in x (product(x_i^-1)). Returns +Inf if x is not positive.

```python
prod_inv(x)
```

--------------------------------

### Equality Constraint with Non-Affine Sides

Source: https://cvxr.com/cvx/doc/_sources/quickstart.txt

Demonstrates an equality constraint where both sides are non-affine, which CVX rejects with an error.

```matlab
norm(x,Inf) == 1;
```

--------------------------------

### Chebyshev Norm Minimization with CVX

Source: https://cvxr.com/cvx/doc/_sources/quickstart.txt

Solves the Chebyshev (infinity) norm minimization problem directly using CVX. CVX handles the underlying transformation automatically.

```matlab
cvx_begin
    variable x(n)
    minimize( norm(A*x-b,Inf) )
cvx_end
```

--------------------------------

### Save CVX Preferences

Source: https://cvxr.com/cvx/doc/solver.html

This command saves the current solver choice and other settings (like cvx_expert, cvx_power_warning, cvx_precision) permanently. Ensure preferences are set correctly before issuing this command.

```matlab
cvx_save_prefs

```

--------------------------------

### Display Current Solver

Source: https://cvxr.com/cvx/doc/_sources/solver.txt

Use this command to check which solver is currently selected in CVX.

```matlab
cvx_solver
```

--------------------------------

### Control CVX Screen Output with cvx_quiet

Source: https://cvxr.com/cvx/doc/solver.html

Use 'cvx_quiet true' to suppress output or 'cvx_quiet false' to restore it. Entering the command with no arguments returns the current setting. Changes made outside a model affect subsequent models.

```matlab
cvx_quiet true

```

```matlab
cvx_quiet false

```

--------------------------------

### Declare Multiple Variables

Source: https://cvxr.com/cvx/doc/basics.html

Use the 'variables' command to declare multiple variables at once. This is a shorthand for declaring several variables with basic properties.

```matlab
variables x1 x2 x3 y1(10) y2(10,10,10);
```

--------------------------------

### Generate Least Squares Data in Matlab

Source: https://cvxr.com/cvx/doc/quickstart.html

Sets up the dimensions and generates random matrices A and b for a least-squares problem.

```matlab
m = 16; n = 8;
A = randn(m,n);
b = randn(m,1);

```

--------------------------------

### Standard LMI Constraint in CVX

Source: https://cvxr.com/cvx/doc/sdp.html

This snippet shows a standard way to express a Hermitian semidefinite constraint in CVX using the `hermitian_semidefinite` function.

```matlab
cvx_begin
    variable Z(n,n) hermitian toeplitz
    dual variable Q
    minimize( norm( Z - P, 'fro' ) )
    Z == hermitian_semidefinite( n ) : Q;
cvx_end

```

--------------------------------

### Corrected Array Assignment with Expression Holder

Source: https://cvxr.com/cvx/doc/basics.html

Shows the correct way to build an array of CVX expressions by first declaring 'x' as an expression holder.

```matlab
variable u(9);
expression x(10);
x(1) = 1;
for k = 1 : 9,
    x(k+1) = sqrt( x(k) + u(k) );
end

```

--------------------------------

### Original Log-Determinant Objective

Source: https://cvxr.com/cvx/doc/_sources/advanced.txt

This represents the standard way a log-determinant maximization problem might be initially formulated.

```matlab
minimize( log_det(X) )
```

--------------------------------

### L1 Norm Minimization with MATLAB linprog

Source: https://cvxr.com/cvx/doc/_sources/quickstart.txt

Solves the L1 norm minimization problem using MATLAB's linprog. This requires reformulating the problem into a linear program.

```matlab
f    = [ zeros(n,1); ones(m,1) ];
Ane  = [ +A, -eye(m); -A, -eye(m) ];
bne  = [ +b; -b ];
xt   = linprog(f,Ane,bne);
x_l1 = xt(1:n,:);
```

--------------------------------

### Equivalent Constraint using geo_mean

Source: https://cvxr.com/cvx/doc/advanced.html

This snippet shows an alternative formulation for a sum_log constraint using the geo_mean function. It is preferred when possible to avoid functions requiring successive approximation.

```matlab
sum_log(x) >= 10

```

```matlab
geo_mean(x) >= 10^(1/length(x))

```

--------------------------------

### Bound-Constrained Least Squares with MATLAB quadprog

Source: https://cvxr.com/cvx/doc/_sources/quickstart.txt

Solves a least-squares problem with upper and lower bounds on x using MATLAB's quadprog. This approach requires transforming the problem into a quadratic program.

```matlab
bnds = randn(n,2);
l = min( bnds, [], 2 );
u = max( bnds, [], 2 );
```

```matlab
x_qp = quadprog( 2*A'*A, -2*A'*b, [], [], [], [], l, u );
```

--------------------------------

### Declare Integer and Binary Variables

Source: https://cvxr.com/cvx/doc/basics.html

Declare variables for Mixed-Integer Convex Programming (MIDCPs). 'integer' restricts the variable to integer values, and 'binary' restricts it to 0 or 1.

```matlab
variable p(10) integer
variable q binary
```

--------------------------------

### Correcting Overdetermined Constraints with Symmetric Part

Source: https://cvxr.com/cvx/doc/_sources/advanced.txt

This snippet shows the corrected formulation for the positive semidefinite problem, using the 'sym' function to ensure the constraint is well-posed when the original matrix W is unsymmetric.

```matlab
sym( W ) + D == semidefinite(n);
```

--------------------------------

### Declare and Associate Dual Variable with Inequality Constraint

Source: https://cvxr.com/cvx/doc/basics.html

Declare a dual variable `y` and associate it with the inequality constraint `A * x <= b` in an LP. This syntax is used to retrieve sensitivity information for the constraint.

```matlab
n = size(A,2);
cvx_begin
    variable x(n);
    dual variable y;
    minimize( c' * x );
    subject to
        y : A * x <= b;
cvx_end

```

--------------------------------

### Define Matrices for Constraints (MATLAB)

Source: https://cvxr.com/cvx/doc/_sources/quickstart.txt

This snippet shows how to define matrices C and d in MATLAB, which will be used for adding constraints to a CVX problem.

```matlab
p = 4;
C = randn(p,n);
d = randn(p,1);
```

--------------------------------

### Using square_pos for Monotonic Compositions

Source: https://cvxr.com/cvx/doc/_sources/dcp.txt

CVX rejects compositions where a non-monotonic function is applied to a convex function. Using monotonic extensions like `square_pos` can resolve this, as shown when the argument is always positive.

```python
square_pos( square( x ) + 1 )
```

--------------------------------

### Quadratic Over Linear

Source: https://cvxr.com/cvx/doc/_sources/funcref.txt

Computes x'x/y for real x and y > 0, or x*x/y for complex x and y > 0. Adds a constraint y > 0 in CVX specification.

```python
quad_over_lin(x,y)
```

--------------------------------

### BibTeX Citation for CVX

Source: https://cvxr.com/cvx/doc/_sources/citing.txt

Use this BibTeX entry when citing the CVX software in academic publications. It includes author, title, how it was published, and the year.

```bibtex
@misc{cvx,
  author       = {CVX Research, Inc.},
  title        = {{CVX}: Matlab Software for Disciplined Convex Programming, version 2.0},
  howpublished = {\url{https://cvxr.com/cvx}},
  month        = aug,
  year         = 2012
}
```

--------------------------------

### CVX Professional License Files

Source: https://cvxr.com/cvx/doc/license.html

These files are governed by the CVX Professional License. Their presence in a package subjects the entire package to this license.

```text
shims/cvx_mosek.p
shims/cvx_gurobi.p
cvx_license.p
```

--------------------------------

### Create Expression Holders

Source: https://cvxr.com/cvx/doc/_sources/funcref.txt

Creates one or more expression holders.

```matlab
expression
```

```matlab
expressions
```

--------------------------------

### Set CVX Solver Settings

Source: https://cvxr.com/cvx/doc/_sources/funcref.txt

Allows the user to deliver advanced, solver-specific settings to the solver that CVX does not otherwise support.

```matlab
cvx_solver_settings
```

--------------------------------

### Clear All Custom Settings for All Solvers

Source: https://cvxr.com/cvx/doc/solver.html

This command removes all custom settings for all solvers. It is a comprehensive way to reset all solver configurations to their default states.

```matlab
cvx_solver_settings -clearall

```

--------------------------------

### Simplified Objective without log_det

Source: https://cvxr.com/cvx/doc/advanced.html

This demonstrates how to simplify a log_det objective by removing the logarithm and solving for the root-n determinant instead. The log_det value can be computed post-optimization.

```matlab
minimize( log_det(X) )

```

```matlab
minimize( n*log(det_rootn(X)) )

```

```matlab
minimize( det_rootn(X) )

```

--------------------------------

### Control Solver Screen Output

Source: https://cvxr.com/cvx/doc/_sources/solver.txt

Toggle screen output for the solver. 'cvx_quiet true' suppresses output, while 'cvx_quiet false' restores it. Commands issued within a cvx_begin/cvx_end block affect only that model.

```matlab
cvx_quiet true
```

```matlab
cvx_quiet false
```