Sequential Quadratic Programming

Abstract

Since its popularization in the late 1970s, Sequential Quadratic Programming (SQP) has arguably become the most successful method for solving nonlinearly constrained optimization problems. As with most optimization methods, SQP is not a single algorithm, but rather a conceptual method from which numerous specific algorithms have evolved. Backed by a solid theoretical and computational foundation, both commercial and public-domain SQP algorithms have been developed and used to solve a remarkably large set of important practical problems. Recently large-scale versions have been devised and tested with promising results.

Semismooth SQP method for equality-constrained optimization problems with an application to the lifted reformulation of mathematical programs with complementarity constraints

“…Given the current iterate (x k , λ k ) ∈ R n ×R l , an iteration of the sequential quadratic programming (SQP) algorithm [2] for problem (1) consists of computing a station-ary point x k+1 and an associated Lagrange multiplier λ k+1 of the subproblem…”

“…When conditions (33) and (34) hold,μ is called an MPCC-multiplier associated to the strongly stationary pointx of problem (2). It is said that the MPCC linear independence constraint qualification (MPCC-LICQ) holds at a feasible pointx if…”

“…We thus have reasons to expect this condition to hold. Theorem 3.2 Let the derivatives of f , G and H be semismooth at a strongly stationary pointx of problem (2), with an associated MPCC-multiplierμ. Let the sequence {(x k , y k , λ k )} ⊂ R n × R l × (R m × R m ) be such that for each k the point (x k+1 , y k+1 , λ k+1 ) satisfies (45), where the matrix H k is of the form (48) with some symmetric n × n-matrix H k , and with a k ∈ R m defined by (49), (47) and (50) for some function ρ :…”

“…The advantage is that the problem (3) has better regularity properties than when the power s = 3 is used; see the discussion in [12] and also Remark 4 below. In this paper, we consider the semismooth version of the sequential quadratic programming method (SQP) [2] for problem (1), and its modification tailored specifically to the structure of lifted MPCC (3). It should be noted that, in local analysis, semismooth SQP for (1) is just the semismooth Newton method (SNM) applied to the Lagrange optimality system of (1).…”

“…This strategy is more in the spirit of classical SQP algorithms [2] and can be advantageous compared to the use of the residual of the Lagrange system. In particular, it is natural to expect that globalization using the penalty function, being more optimization-oriented, should outperform globalization based on the residual in terms of the "quality" of the output: the former should be much less attracted by nonoptimal stationary points than the latter.…”