Unlocking Optimization Success: Why Solver Choice Matters

When it comes to mathematical optimization, choosing the right solver is not just a technicality—it’s the key to unlocking efficiency, accuracy, and performance. Whether you’re tackling a sparse or dense problem, the solver you select profoundly impacts the resources you’ll need and the results you’ll achieve. This blog dives into the critical decision-making process of solver selection, why it matters, and how you can avoid common pitfalls.

Why Optimization Solver Choice is Essential

Optimization problems are as diverse as the industries they serve, from structural engineering to data science. But at their core, they all share a common challenge: efficiently balancing the computational cost with the scale and complexity of the problem. One of the most important questions in this process is: Is your problem sparse or dense? Answering this question early enables you to align your solver choice with the unique structure of your problem, saving time and computational resources.

Sparse vs. Dense Problems: Why It Matters

• Sparse Problems: These involve matrices where most elements are zero. Sparse solvers leverage this structure to avoid unnecessary computations, dramatically reducing memory and processing time.
• Dense Problems: These have many interdependencies among variables, requiring solvers designed to handle large, fully populated matrices efficiently.

The Hidden Value of Sparsity

For large-scale optimization, recognizing and exploiting sparsity can transform performance. Consider this example:

• A matrix with thousands of variables but a density below 1% enables sparse solvers to skip over zeros, cutting computational time and memory use significantly.
• Dense solvers, in contrast, struggle as problem size grows, often consuming exponentially more resources.

The results? Sparse solvers consistently outperform dense counterparts for problems with low-density structures.

Real-World Applications: Lessons Learned

• Linear Programming: For problems with densities below 1%, sparse solvers can unlock dramatic performance gains. Sparse solvers complete tasks in seconds where dense solvers take minutes or hours – note not all linear programming problems are sparse.
• Cholesky Factorization: Sparse Cholesky routines scale seamlessly for large problems, while dense methods hit performance walls as problem size and complexity increase.

The Danger of Misaligned Solvers

Even experienced users can misinterpret their problem’s structure. It’s not uncommon to overlook the sparsity introduced during problem reformulation – see the example in this blog.

Key Takeaways for Optimal Solver Choice

• Assess Sparsity Early: Determine the density of your problem’s matrices and choose a solver that aligns with its structure.
• Consider Reformulations: The data your solver processes may differ from the original problem. Analyse what the solver actually sees.
• Leverage Dedicated Solvers: When available, they provide unparalleled efficiency by tailoring methods to your specific problem type.

Empower Your Optimization Workflow

Choosing the right solver isn’t just about saving time—it’s about transforming the way you approach optimization. By aligning your solver choice with the problem’s structure, you’ll achieve faster, more efficient, and scalable solutions. Sign up to receive more mathematical optimization resources.

Robust, Tested, Performant Optimization Solvers

The Optimization Modelling Suite – delivered with the nAG Library – features an extensive collection of Mathematical Optimization solvers. The solvers are accessed via an intuitive interface designed for ease of use. Key mathematical optimization areas covered include:

• Linear Programming (LP) – dense and sparse, based on simplex method and interior point method;
• Quadratic Programming (QP) – convex and nonconvex, dense and sparse;
• Second-order Cone Programming (SOCP) – covering many convex optimization problems, such as Quadratically Constrained Quadratic Programming (QCQP);
• Nonlinear Programming (NLP) – dense and sparse, based on active-set SQP methods and interior point method (IPM);
• Global Nonlinear Programming – algorithms based on multistart, branching, and metaheuristics;
• Mixed Integer Linear Programming (MILP) – for large-scale problems, based on a modern branch-and-bound approach;
• Mixed Integer Nonlinear Programming (MINLP) – for dense (possibly nonconvex) problems;
• Semidefinite Programming (SDP) – both linear matrix inequalities (LMI) and bilinear matrix inequalities (BMI);
• Derivative-free Optimization (DFO) – solvers for problems where derivatives are unavailable and approximations are inaccurate;
• Least Squares (LSQ), data fitting, calibration, regression – linear and nonlinear, constrained and unconstrained.

View nAG’s optimization solver documentation.