Defining OptimizationProblems

Defines an optimization problem. Documentation Page: https://docs.sciml.ai/Optimization/stable/API/optimization_problem/

Mathematical Specification of an Optimization Problem

To define an optimization problem, you need the objective function $f$ which is minimized over the domain of $u$, the collection of optimization variables:

\[min_u f(u,p)\]

$u₀$ is an initial guess for the minimizer. f should be specified as f(u,p) and u₀ should be an AbstractArray whose geometry matches the desired geometry of u. Note that we are not limited to vectors for u₀; one is allowed to provide u₀ as arbitrary matrices / higher-dimension tensors as well.

Problem Type

Constructors

OptimizationProblem{iip}(f, u0, p = SciMLBase.NullParameters(),;
                        lb = nothing,
                        ub = nothing,
                        lcons = nothing,
                        ucons = nothing,
                        sense = nothing,
                        kwargs...)

isinplace optionally sets whether the function is in-place or not. This is determined automatically, but not inferred. Note that for OptimizationProblem, in-place refers to the objective's derivative functions, the constraint function and its derivatives. OptimizationProblem currently only supports in-place.

Parameters p are optional, and if not given, then a NullParameters() singleton will be used, which will throw nice errors if you try to index non-existent parameters.

lb and ub are the upper and lower bounds for box constraints on the optimization variables. They should be an AbstractArray matching the geometry of u, where (lb[i],ub[i]) is the box constraint (lower and upper bounds) for u[i].

lcons and ucons are the upper and lower bounds in case of inequality constraints on the optimization and if they are set to be equal then it represents an equality constraint. They should be an AbstractArray, where (lcons[i],ucons[i]) are the lower and upper bounds for cons[i].

The f in the OptimizationProblem should typically be an instance of OptimizationFunction to specify the objective function and its derivatives either by passing predefined functions for them or automatically generated using the ADType.

If f is a standard Julia function, it is automatically transformed into an OptimizationFunction with NoAD(), meaning the derivative functions are not automatically generated.

Any extra keyword arguments are captured to be sent to the optimizers.

Fields

• f: the function in the problem.
• u0: the initial guess for the optimization variables.
• p: Either the constant parameters or fixed data (full batch) used in the objective or a MLUtilsDataLoader for minibatching with stochastic optimization solvers. Defaults to NullParameters.
• lb: the lower bounds for the optimization variables u.
• ub: the upper bounds for the optimization variables u.
• int: integrality indicator for u. If int[i] == true, then u[i] is an integer variable. Defaults to nothing, implying no integrality constraints.
• lcons: the vector of lower bounds for the constraints passed to OptimizationFunction. Defaults to nothing, implying no lower bounds for the constraints (i.e. the constraint bound is -Inf)
• ucons: the vector of upper bounds for the constraints passed to OptimizationFunction. Defaults to nothing, implying no upper bounds for the constraints (i.e. the constraint bound is Inf)
• sense: the objective sense, can take MaxSense or MinSense from Optimization.jl.
• kwargs: the keyword arguments passed on to the solvers.

Inequality and Equality Constraints

Both inequality and equality constraints are defined by the f.cons function in the OptimizationFunction description of the problem structure. This f.cons is given as a function f.cons(u,p) which computes the value of the constraints at u. For example, take f.cons(u,p) = u[1] - u[2]. With these definitions, lcons and ucons define the bounds on the constraint that the solvers try to satisfy. If lcons and ucons are nothing, then there are no constraints bounds, meaning that the constraint is satisfied when -Inf < f.cons < Inf (which of course is always!). If lcons[i] = ucons[i] = 0, then the constraint is satisfied when f.cons(u,p)[i] = 0, and so this implies the equality constraint u[1] = u[2]. If lcons[i] = ucons[i] = a, then $u[1] - u[2] = a$ is the equality constraint.

Inequality constraints are then given by making lcons[i] != ucons[i]. For example, lcons[i] = -Inf and ucons[i] = 0 would imply the inequality constraint $u[1] <= u[2]$ since any f.cons[i] <= 0 satisfies the constraint. Similarly, lcons[i] = -1 and ucons[i] = 1 would imply that -1 <= f.cons[i] <= 1 is required or $-1 <= u[1] - u[2] <= 1$.

Note that these vectors must be sized to match the number of constraints, with one set of conditions for each constraint.

Data handling

As described above the second argument of the objective definition can take a full batch or a DataLoader object for mini-batching which is useful for stochastic optimization solvers. Thus the data either as an Array or a DataLoader object should be passed as the third argument of the OptimizationProblem constructor. For an example of how to use this data handling, see the Sophia example in the Optimization.jl documentation or the mini-batching tutorial.