Solvers Ecosystem

The solver ecosystem in JSO is in constant development. We are looking into unifying the solver structure to make solvers, subsolvers, and tools plug-and-play.

Within JSO, our solver development has focused on three types of packages:

• New methods, usually accompanied by research papers;

• our implementation of (well- or no-so-well-)known methods;

• wrappers around noteworthy solvers implemented in other languages.

Solver list

Below is the list of solvers and the type of problems that they are designed to solve.

Solver	Package	Description	Types of problem
cannoles	CaNNOLeS.jl	Regularization method for nonlinear least squares	Equality-constrained NLS
dci	DCISolver.jl	Trust-cylinder similar to two-step SQP	Equality-constrained NLP
lbfgs	JSOSolvers.jl	Factorization-free linesearch limited-memory inverse BFGS	Unconstrained NLP
percival	Percival.jl	Factorization-free augmented Lagrangian	Generally-constrained NLP
ripqp	RipQP.jl	Regularized Interior-Point Quadratic Programming	Convex QP (incl. LP)
tron	JSOSolvers.jl	Factorization-free trust-region second-order or quasi-Newton method	Bound-constrained NLP
tronls	JSOSolvers.jl	Least-squares versions of tron	Unconstrained NLS
trunk	JSOSolvers.jl	Factorization-free trust-region second-order or quasi-Newton method	Unconstrained NLP
trunkls	JSOSolvers.jl	Least-squares version of trunk	Unconstrained NLS

Types of problems

Our optimization problems can be defined as

$$\text{minimize} \quad f(x) \quad \text{subject to} \quad x \in \Omega.$$

Our constraint types and their meanings are:

• Unconstrained: $\Omega = \mathbb R^n$.

• Bound constrained: $\Omega = \{x \in \mathbb R^n |\ \ell \leq x \leq u\}$.

• Equality constrained: $\Omega = \{x \in \mathbb R^n |\ c(x) = 0\}$.

• Generally constrained: $\Omega = \{x \in \mathbb R^n |\ c_L \leq c(x) \leq c_U \}$, where $c_{L_i} = c_{U_i}$ is a valid possibility, as are $c_{L_i} = -\infty$ and $c_{U_i} = \infty$, but not at the same time.

In addition to differences in the constraints, we also have different objective types:

• NLP: General objective. The objective $f(x)$ of these problems has no special structure to exploit. Access to its derivatives occurs through the default NLPModels.jl API.

• NLS: Nonlinear Least-Squares problems. These problems are defined by $f(x) = \tfrac{1}{2}\Vert F(x)\Vert^2$, with $F$ and its derivatives available through the API for NLSModels, part of NLPModels.jl

• QP: Quadratic Programming. These problems are defined by $f(x) = \tfrac{1}{2}x^T Q x + g^T x + c$, with the API defined by QuadraticModels.jl.