Primal-dual interior methods for nonlinear programs

Describe inequalities by g(x)≤0 and introduce positive slacks s so g(x)+s=0. Their multipliers z are also positive during interior iterations. With equality multipliers λ unrestricted, stationarity is the gradient of f plus the equality and inequality Jacobian-transpose actions. Exact complementarity would require each sᵢzᵢ to vanish. Replace that product by a positive barrier parameter τ to obtain a smooth system inside the positive orthant, then reduce τ as the solve advances.

A feasible barrier formulation minimizes f minus τ times the sum of log slacks, with the corresponding equality equations. Its stationary equations give zᵢ=τ/sᵢ, matching perturbed complementarity. Infeasible-start primal-dual methods instead permit nonzero equality and slack-equation residuals while maintaining positivity of s and z. Thus interior does not necessarily mean every nonlinear inequality at the current x is already satisfied; it can refer to the algorithmic slack and multiplier variables.

Linearizing the residuals produces a coupled Newton system containing the Lagrangian Hessian, constraint Jacobians, and slack-multiplier diagonal blocks. A full step can make a positive slack or multiplier negative. A fraction-to-the-boundary rule limits the step before that happens, commonly retaining a chosen fraction of the distance to zero. This positivity bound is only one acceptance condition: a filter or merit test must also judge nonlinear residual and objective progress.

In nonconvex problems, eliminating variables can produce indefinite or nearly singular systems. Inertia checks and regularization help ensure that the chosen linear system has the intended curvature behavior. If the algorithm cannot make useful progress toward feasibility, a restoration phase may solve a different subproblem. These features are not cosmetic implementation details: their assumptions and safeguards are part of a convergence theory. Merely taking Newton steps on the perturbed equations can converge to an unsuitable stationary branch or fail altogether.

A small barrier parameter alone is not a stopping certificate. Monitor original stationarity, equality feasibility, inequality feasibility, multiplier nonnegativity, and complementarity on scales appropriate to the application. A point with small products but a large primal residual can be far from a valid constrained solution. Likewise, positive slacks at a nonzero τ describe an approximation to an active boundary, not a failure to understand where the exact optimum is located.

Under strong regularity conditions near a KKT solution, including suitable constraint qualifications, second-order sufficiency, and often strict complementarity in simple path theorems, a local smooth branch and rapid Newton behavior can be established. Degenerate cases need separate analysis. General nonconvex programs do not inherit the global optimality certificates or self-concordant barrier complexity of convex conic problems. Report which residual was bounded and which hypotheses supported the claim, rather than calling every small-τ answer globally optimal.