A data‐science team is tuning a pricing engine whose objective is twice-differentiable and non-convex, subject to hundreds of inequality constraints and simple bounds. They have analytic gradients and Hessians and want every iterate to remain strictly inside the feasible region throughout the search. To do this, they choose a solver that
augments the objective with a logarithmic barrier term −μ ∑log sᵢ(x) to prevent boundary violations,
follows a central path by gradually decreasing the barrier parameter μ→0, and
at each outer iteration solves a primal-dual Newton system instead of a quadratic programming subproblem.
Which class of constrained nonlinear optimization algorithms matches this strategy?
The described approach is characteristic of primal-dual interior-point (path-following) methods. Interior-point algorithms embed all inequality constraints in a logarithmic barrier, ensuring every iterate remains strictly feasible; they then trace the central path toward optimality while repeatedly updating μ and solving Newton-type systems.
Sequential quadratic programming also handles nonlinear constraints, but it linearizes the constraints and solves a series of quadratic programming subproblems rather than inserting a barrier term.
Augmented Lagrangian methods add a quadratic penalty to the Lagrangian and relax feasibility between outer iterations; iterates can lie outside the feasible region.
The Nelder-Mead simplex search is a derivative-free direct-search technique that is typically used for unconstrained (or only box-constrained) problems and does not rely on gradients, Hessians, or barrier terms.
Therefore, only the interior-point class fits all three bullet-point requirements.
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What is a primal-dual interior-point method?
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Why is the logarithmic barrier term −μ ∑log sᵢ(x) used?
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How does a primal-dual Newton system work in this context?