Sequential quadratic programming

For equalities h=0 and inequalities g≤0, construct a quadratic in the displacement with gradient equal to the current objective gradient and matrix B approximating the Hessian of the Lagrangian. Impose the linearized equalities and inequalities on that displacement. Using only the objective Hessian can miss important curvature from active constraints. The QP multipliers also supply estimates for the nonlinear problem's multipliers, while a line search or trust-region strategy decides how much of the computed displacement to use.

For equality constraints, differentiating the KKT stationarity and feasibility equations yields a saddle-point Newton system. With an exact Lagrangian Hessian, the QP step and its updated multiplier solve that same system. This connection explains why SQP can be locally fast. It also explains why a poorly conditioned constraint Jacobian or an incorrect multiplier sign can damage both primal motion and dual estimates. The notation below uses row Jacobians, so the transpose in stationarity is essential.

A linearized equality removes its first-order residual but generally leaves a second-order nonlinear residual after the step. Large tangent motions can therefore move far away from the true feasible set. A merit function combines objective and constraint violation with a chosen penalty scale, whereas a filter treats them as separate progress measures with safeguards. Neither device should be reduced to accepting any objective decrease: that would permit excellent objective values at useless, highly infeasible points.

The linearized QP may itself be infeasible, especially far from a solution or when constraint gradients are degenerate. Elastic variables, feasibility-restoration phases, or trust-region modifications can recover a meaningful next task. Another subtlety is the Maratos effect: even near a regular solution, a full fast local step can be rejected by some merit-function searches because the second-order constraint error is penalized too strongly. A second-order correction can address that specific mismatch without replacing the entire SQP framework.

For an equality-constrained problem with sufficiently smooth derivatives, full-row-rank constraint Jacobian, and positive-definite reduced Lagrangian Hessian at a KKT solution, the KKT Jacobian is nonsingular. If its derivative is locally Lipschitz, exact Newton/SQP steps from sufficiently close primal and multiplier estimates converge quadratically. This is a local result for exact systems and accepted full steps. Inequality versions need careful active-set regularity, often including strict complementarity in simple statements; they cannot be imported without their extra assumptions.

Quasi-Newton SQP replaces the exact curvature with updates, so its convergence rate requires separate approximation conditions. Globalization also changes the sequence away from the solution. Evaluate a solver using primal residuals, Lagrangian gradient residuals, complementarity, and accepted step lengths rather than objective values alone. In a design problem, returning an inexpensive design that violates a safety equality is a failure even if the optimizer's internal quadratic program was solved to machine precision.