Optimization Algorithms FAQ: Simplex, Barrier, and MIP Methods

An optimization algorithm is the procedure used to search for a best feasible solution and, when possible, prove it is optimal. The model type (LP, QP, MIP, and so on) describes the math structure you wrote down; the algorithm describes how the solver works through that structure to return a solution status, objective value, and diagnostics. 

For linear programming (LP), simplex and barrier are two common choices. In business terms, you care about (a) time to a solution that meets your planning deadline and (b) what post-solve insight you need.

Simplex is often a strong fit when you want interpretable sensitivity signals (like dual values) for what-if analysis in pricing, blending, or network flow. Barrier can be attractive for very large, sparse LPs where getting a high-quality continuous solution quickly is the priority, such as multi-period supply planning or large network design relaxations. There is no universal winner; structure and scaling matter. 

Usually, no. In most deployments, customers rely on Gurobi Optimizer defaults because they are designed to work well across a wide range of model types and structures.

Practically, this means you can focus on getting the model formulation and data right, then use solver outputs (status, runtime, incumbent, best bound, and optimality gap) to decide whether the result meets your operational requirements. Manual algorithm choices can be useful in targeted situations, such as benchmarking alternative approaches on a specific model family, or when you have repeatable evidence that one approach is consistently better for your structure.

Even then, treat it as an optimization engineering decision backed by measurements, not a requirement for getting good results. 

Barrier (interior-point) methods are especially relevant for convex quadratic programming and convex quadratic constraints (QP and QCP), and often for conic formulations many teams use for risk or engineering approximations.

Examples include portfolio-style risk models with quadratic variance, or energy and process models that naturally produce convex quadratic relationships. For convex problems solved to completion, a solver can provide a proven optimal solution (or prove infeasibility or unboundedness). 

Mixed-integer programming (MIP) is how you represent discrete decisions: open or close a facility, assign a job to one machine, choose a shift pattern, pick a set of projects under a budget, or enforce on-off logic. Branch-and-bound is the core framework that lets you make those discrete choices while still using LP or QP relaxations to guide the search. Operationally, this matters because you can stop early and still get a feasible plan plus a bound that helps you quantify how far that plan could be from the true optimum. 

Many modern MIP solves add problem-specific tightening steps that improve bounds and reduce search effort. You do not usually manage these details directly; you experience them as better progress on the best bound and, often, faster closure of the optimality gap. The practical takeaway is to watch the solve log signals (incumbent, best bound, gap, time) rather than fixating on a single named method. 

If a MIP solve finishes, you get a proven optimal solution. If you stop early (time limit, node limit, or other stopping criteria), you get the best feasible solution found (the incumbent) and a best bound. The relative optimality gap summarizes the remaining uncertainty between those two values. In applications like last-mile routing, shift scheduling, or daily production planning, a small gap can be enough to deploy confidently, while a large gap may trigger a fallback policy (extend runtime, simplify constraints, or accept a heuristic plan). The right threshold is a business decision tied to KPIs and risk tolerance, not a fixed number. 

Nonconvex quadratic terms show up in bilinear relationships (price x demand, blending interactions, pooling-like mixing effects) and in some engineering approximations. These problems can have multiple local optima, so solvers use global-optimization frameworks (often called spatial branch-and-bound) to search for a globally optimal solution or provide the best incumbent with a bound if stopped early. Expect runtime to be more sensitive to formulation choices and bounds on variables, and plan for scenario testing rather than assuming a single run will always close the gap. 

Yes. In tight decision windows (same-day scheduling, replanning after disruptions, interactive scenario analysis), the practical goal is often a feasible solution quickly, then steady improvement. Heuristics help find good incumbents early, which is valuable even if you later let the solver work on proving optimality. If you manually adjust a solution outside the solver, you can violate constraints, so recheck feasibility or re-optimize before deploying. 

Algorithm performance depends heavily on what you feed it. Common drivers include: 

• Overly large big-M constants that weaken bounds in MIP models 

• Poor scaling (coefficients with extreme magnitudes) that can cause numerical trouble 

• Missing or loose variable bounds, especially in nonconvex quadratic models 

• Symmetry or redundant choices that create many equivalent solutions 

Treat data validation and model governance as part of optimization quality assurance: log solve status, gap, runtime, and infeasibility diagnoses to catch regressions.