# Getting Started

This user guide describes the basic structures and functions to define an optimisation problem, to solve the problem and to analyse the result. If you want to use JuMP to describe the problem, see the JuMP Interface section.

COSMO solves optimisation problems in the following format:

$$$\begin{array}{ll} \text{minimize} & \textstyle{\frac{1}{2}}x^\top Px + q^\top x\\ \text{subject to} & Ax + s = b \\ & s \in \mathcal{K}, \end{array}$$$

with decision variables $x \in \mathbb{R}^n$, $s \in \mathbb{R}^m$ and data matrices $P=P^\top \succeq 0$, $q \in \mathbb{R}^n$, $A \in \mathbb{R}^{m \times n}$, and $b \in \mathbb{R}^m$. The convex set $\mathcal{K}$ is a composition of convex sets and cones.

## Model

The problem data, user settings and workspace variables are all stored in a Model. To get started define an empty model:

model = COSMO.Model()

To initialize the model with an optimisation problem we need to define three more things:

• the objective function, i.e. the matrix P and the vector q in $\frac{1}{2}x^\top P x + q^\top x$
• an array of constraints
• a Settings object that specifies how COSMO solves the problem (optional)

## Objective Function

To set the objective function of your optimisation problem simply define the square positive semidefinite matrix $P \in \mathrm{R}^{n\times n}$ and the vector $q \in \mathrm{R}^{n}$. You might have to transform your optimisation problem into a solver compatible format for this step.

## Constraints

The COSMO interface expects constraints to have the form $A_i x + b_i \in \mathcal{K}_i$, where $\mathcal{K}_i$ is one of the convex sets defined below:

Convex SetDescription
ZeroSetThe set $\{ 0 \}^{dim}$ that contains the origin
NonnegativesThe nonnegative orthant $\{ x \in \mathbb{R}^{dim} : x_i \ge 0, \forall i=1,\dots,\mathrm{dim} \}$
Box(l, u)The hyperbox $\{ x \in \mathbb{R}^{dim} : l \leq x \leq u\}$ with vectors $l \in \mathbb{R}^{dim} \cup \{-\infty\}$ and $u \in \mathbb{R}^{dim} \cup \{+\infty\}$
SecondOrderConeThe second-order (Lorenz) cone $\{ (t,x) \in \mathbb{R}^{dim} : |x|_2 \leq t \}$
PsdConeThe vectorized positive semidefinite cone $\mathcal{S}_+^{dim}$. $x$ is the vector obtained by stacking the columns of the positive semidefinite matrix $X$, i.e. $X \in \mathbb{S}^{\sqrt{dim}}_+ \rarr \text{vec}(X) = x \in \mathcal{S}_+^{dim}$
PsdConeTriangleThe vectorized positive semidefinite cone $\mathcal{S}_+^{dim}$. $x$ is the vector obtained by stacking the columns of the upper triangular part of the positive semidefinite matrix $X$, i.e. $X \in \mathbb{S}^{d}_+ \rarr \text{svec}(X) = x \in \mathcal{S}_+^{dim}$ where $d=\sqrt{1/4 + 2 \text{dim}} - 1/2$
ExponentialConeThe exponential cone $\mathcal{K}_{exp} = \{(x, y, z) \mid y \geq 0, ye^{x/y} ≤ z\} \cup \{ (x,y,z) \mid x \leq 0, y = 0, z \geq 0 \}$
DualExponentialConeThe dual exponential cone $\mathcal{K}^*_{exp} = \{(x, y, z) \mid x < 0, -xe^{y/x} \leq e^1 z \} \cup \{ (0,y,z) \mid y \geq 0, z \geq 0 \}$
PowerCone(alpha)The 3d power cone $\mathcal{K}_{pow} = \{(x, y, z) \mid x^\alpha y^{(1-\alpha)} \geq |z|, x \geq 0, y \geq 0 \}$ with $0 < \alpha < 1$
DualPowerCone(alpha)The 3d dual power cone $\mathcal{K}^*_{pow} = \{(u, v, w) \mid \left( \frac{u}{\alpha}\right)^\alpha \left( \frac{v}{1-\alpha}\right)^{(1-\alpha)} \geq |w|, u \geq 0, v \geq 0 \}$ with $0 < \alpha < 1$

The constructor for a constraint expects a matrix A, a vector b and a convex_set.

Lets consider a problem with a decision variable $x \in \mathbb{R}^5$. Suppose we want to create the two constraint $x_2 + 5 \geq 0$ and $x_3 - 3 \geq 0$. We can do this either by creating two constraints and adding them to an array:

  constraint1 = COSMO.Constraint([0.0 1.0 0.0 0.0 0.0], 5.0, COSMO.Nonnegatives)
constraint2 = COSMO.Constraint([0.0 0.0 1.0 0.0 0.0], -3.0, COSMO.Nonnegatives)
constraints = [constraint1; constraint2]

The second option is to include both in one constraint:

constraint1 = COSMO.Constraint([0.0 1.0 0.0 0.0 0.0; 0.0 0.0 1.0 0.0 0.0], [5.0; -3.0], COSMO.Nonnegatives)

Another way to construct the constraint is to used the optional arguments dim, the dimension of x, and indices, the elements of x that appear in the constraint. When specifying these arguments, A and b only refer to the elements of x in indices:

constraint1 = COSMO.Constraint([1.0 0.0; 0.0 1.0], [5.0; -3.0], COSMO.Nonnegatives, 5, 2:3)

Consider as a second example the positive semidefinite constraint on a matrix $X \in \mathbb{S}_+^{3}$. Our decision variable is the vector $x$ obtained by stacking the columns of $X$. We can specify the constraint on $x$ in the following way:

$$$I_9 x + \{0\}_9 \in \mathcal{S}_+^9,$$$

or in Julia:

constraint1 = COSMO.Constraint(Matrix(1.0I, 9, 9), zeros(9), COSMO.PsdCone)

Several constraints can be combined in an array:

constraints = [constraint_1, constraint_2, ..., constraint_N]

It is usually enough to pass the convex_set as a type. However, some convex sets like Box, PowerCone and DualPowerCone require more information to be created. In that case you have to pass an object to the constructor, e.g.

l = -ones(2)
u = ones(2)
constraint = COSMO.Constraint(Matrix(1.0I, 2, 2), zeros(2), COSMO.Box(l, u))

or in the case of a power Cone you specify the alpha:

constraint = COSMO.Constraint(Matrix(1.0I, 3, 3), zeros(3), COSMO.PowerCone(0.6))

## Settings

The solver settings are stored in a Settings object and can be adjusted by the user. To create a Settings object just call the constructor:

COSMO.SettingsType
COSMO.Settings{T}(; kwargs) where {T <: AbstractFloat}

Creates a COSMO settings object that is used to pass user settings to the solver.

ArgumentDescriptionValues (default)
alphaRelaxation parameter1.6
eps_absAbsolute residual tolerance1e-5
eps_relRelative residual tolerance1e-5
eps_prim_infPrimal infeasibility tolerance1e-5
eps_dual_infDual infeasibility tolerance1e-5
max_iterMaximum number of iterations5000
verboseVerbose printingfalse
verbose_timingVerbose timingfalse
kkt_solverLinear System solverQdldlKKTSolver
check_terminationCheck termination interval25
check_infeasibilityCheck infeasibility interval40
scalingNumber of scaling iterations10
decomposeActivate to decompose chordal psd constraintstrue
complete_dualActivate to complete the dual variable after decompositionfalse
merge_strategyChoose a strategy for clique mergingCliqueGraphMerge
compact_transformationChoose how a decomposed problem is transformedtrue
time_limitSet solver time limit in s0 (deactivated)
acceleratorAcceleration schemeAndersonAccelerator{T, Type2{QRDecomp}, RestartedMemory, NoRegularizer}
accelerator_activationAccelerator activationImmediateActivation
safeguardAccelerator safeguardingtrue
safeguard_tolSafeguarding tolerance2.0
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To adjust those values, either pass your preferred option and parameter as a key-value pair to the constructor or edit the corresponding field afterwards. For example if you want to enable verbose printing and increase the solver accuracy, you can type

settings = COSMO.Settings(verbose = true, eps_abs = 1e-5, eps_rel = 1e-5)
# the following is equivalent
settings = COSMO.Settings()
settings.verbose = true
settings.eps_abs = 1e-5
settings.eps_rel = 1e-5

## Assembling the model

Once the objective function and an array of constraints have been defined, we can assemble the model with

COSMO.assemble!(model, P, q, constraints)

This simply sets the corresponding variables in the model and transforms the array of constraints into the problem format defined at the top of the page.

If you want to change the default settings, you can pass your settings object custom_settings to the assemble! function:

COSMO.assemble!(model, P, q, constraints, settings = custom_settings)

## Warm starting

One of the advantages of ADMM-based solvers is that they can be easily warm started. By providing starting values for the primal variable x and/or the dual variable y in the vicinity of their optimal values, the number of iterations to convergence can often be dramatically decreased.

Consider the case where you have a decision variable $x \in \mathbb{R}^3$ and a dual variable $y \in \mathbb{R}^2$. Assume you expect their optimal values to be close to $x_0 = (1, 5, 3)$ and $y_0 = (1, 2)$. You can pass these values when assembling the model.

x_0 = [1.0; 5.0; 3.0]
y_0 = [1.0; 2.0]
COSMO.assemble!(model, P, q, constraints, x0 = x_0, y0 = y_0)

Another option is to use

COSMO.assemble!(model, P, q, constraints)
warm_start_primal!(model, x_0)
warm_start_dual!(model, y_0)

## Solving

After the model has been assembled, we can solve the problem by typing

results = COSMO.optimize!(model)

Once the solver algorithm terminates, it will return a Results object that gives information about the status of the solver. If successful, it contains the optimal objective value and optimal primal and dual variables. For more information see the following section.

## Results

After attempting to solve the problem, COSMO will return a result object with the following fields:

COSMO.ResultType
Result{T <: AbstractFloat}

Object returned by the COSMO solver after calling optimize!(model). It has the following fields:

FieldnameTypeDescription
xVector{T}Primal variable
yVector{T}Dual variable
sVector{T}(Primal) set variable
obj_valTObjective value
iterIntTotal number of ADMM iterations (incl. safeguarding_iter)
safeguarding_iterIntNumber of iterations due to safeguarding of accelerator
statusSymbolSolution status
timesCOSMO.ResultTimesStruct with several measured times
source
COSMO.ResultInfoType
ResultInfo{T <: AbstractFloat}

Object that contains further information about the primal residual, the dual residuals and the rho updates.

source

### Status Codes

COSMO will return one of the following statuses:

Status CodeDescription
:SolvedAn optimal solution was found
:UnsolvedDefault value
:Max_iter_reachedSolver reached iteration limit (set with Settings.max_iter)
:Time_limit_reachedSolver reached time limit (set with Settings.time_limit)
:Primal_infeasibleProblem is primal infeasible
:Dual_infeasibleProblem is dual infeasible

### Timings

If settings.verbose_timing is set to true, COSMO will report the following times in result.times:

COSMO.ResultTimesType
ResultTimes

Part of the Result object returned by the solver. ResultTimes contains timing results for certain parts of the algorithm:

Time NameDescription
solver_timeTotal time used to solve the problem
setup_timeSetup time = graph_time + init_factor_time + scaling_time
scaling_timeTime to scale the problem data
graph_timeTime used to perform chordal decomposition
init_factor_timeTime used for initial factorisation of the system of linear equations
factor_update_timeSum of times used to refactor the system of linear equations due to rho
iter_timeTime spent in iteration loop
proj_timeTime spent in projection functions
post_timeTime used for post processing
update_timeTime spent in the update! function of the accelerator
accelerate_timeTime spent in the accelerate! function of the accelerator

By default COSMO only measures solver_time, setup_time and proj_time. To measure the other times set verbose_timing = true.

source

It holds: solver_time = setup_time+ iter_time + factor_update_time + post_time,

setup_time = graph_time+ init_factor_time + scaling_time,

proj_time is a subset of iter_time.

## Updating the model

In some cases we want to solve a large number of similar models. COSMO allows you to update the model problem data vectors q and b after the first call of optimize!(). After changing the problem data, COSMO can reuse the factorisation step of the KKT matrix from the previous problem which can save a lot of time in the case of LPs and QPs.

COSMO.update!Function
update!(model, q = nothing, b = nothing)

Updates the model data for b or q. This can be done without refactoring the KKT matrix. The vectors will be appropriatly scaled.

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