## A Computationally Efficient Feasible Sequential Quadratic Programming Algorithm (2001)

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Venue: | SIAM Journal on Optimization |

Citations: | 23 - 0 self |

### BibTeX

@ARTICLE{Lawrence01acomputationally,

author = {Craig T. Lawrence and André L. Tits},

title = {A Computationally Efficient Feasible Sequential Quadratic Programming Algorithm},

journal = {SIAM Journal on Optimization},

year = {2001},

volume = {11},

pages = {1092--1118}

}

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### Abstract

. A sequential quadratic programming (SQP) algorithm generating feasible iterates is described and analyzed. What distinguishes this algorithm from previous feasible SQP algorithms proposed by various authors is a reduction in the amount of computation required to generate a new iterate while the proposed scheme still enjoys the same global and fast local convergence properties. A preliminary implementation has been tested and some promising numerical results are reported. Key words. sequential quadratic programming, SQP, feasible iterates, feasible SQP, FSQP AMS subject classifications. 49M37, 65K05, 65K10, 90C30, 90C53 PII. S1052623498344562 1.

### Citations

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Citation Context ...tilting parameter starts out positive and asymptotically approaches zero. Recently there has been a great deal of interest in interior point algorithms for nonconvex nonlinear programming (see, e.g., =-=[5, 6, 24, 4, 16, 23]-=-). Such algorithms generate feasible iterates and typically only require the solution of linear systems of equations in order to generate new iterates. SQP-type algorithms, however, are often at an ad... |

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Citation Context ...ions 2--3]. Finally, note that Q-superlinear convergence would follows if Assumption 6 were replaced with the stronger assumption lim k!1 kP k (H k \Gamma r 2 xx L(x ;s))d k k kd k k = 0: (See, e.g., =-=[2]-=-.) 27 4 Implementation and Numerical Results Our implementation of FSQP 0 (in C) differs in a number of ways from the algorithm stated in Section 2. (It is readily checked that none of the differences... |

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Citation Context ...f the algorithm which was intentionally left vague in Sections 2 and 3 was the updating scheme for the Hessian estimates H k . In the implementation, we use the BFGS update with Powell's modification =-=[19]-=-. Specifically, define ffi k+1 \Delta = x k+1 \Gamma x k y k+1 \Delta = r x L(x k+1 ;sk ) \Gamma r x L(x k ;sk ); 30 where, in an attempt to better approximate the true multipliers, ifsk ? p ffl m we ... |

102 | A trust region method based on interior point techniques for nonlinear programming
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Citation Context ...tilting parameter starts out positive and asymptotically approaches zero. There has been a great deal of interest recently in interior point algorithms for nonconvex nonlinear programming (see, e.g., =-=[5, 6, 26, 4, 18, 25]-=-). Such algorithms generate feasible iterates and typically require only the solution of linear systems of equations in order to generate new iterates. SQP-type algorithms, however, are often at an ad... |

91 |
Computational Methods in Optimization
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Citation Context ...for solving (P ). For an excellent recent survey of SQP algorithms, and the theory behind them, see [2]. Denote the feasible set for (P ) by X \Delta = f x 2 R n j g j (x)s0; j = 1; : : : ; m g: 1 In =-=[17, 8, 14, 15, 1], variatio-=-ns on the standard SQP iteration for solving (P ) are proposed which generate iterates lying within X. Such methods are sometimes referred to as "Feasible SQP" (or FSQP) algorithms. It was o... |

82 |
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Citation Context ...tilting parameter starts out positive and asymptotically approaches zero. There has been a great deal of interest recently in interior point algorithms for nonconvex nonlinear programming (see, e.g., =-=[5, 6, 26, 4, 18, 25]-=-). Such algorithms generate feasible iterates and typically require only the solution of linear systems of equations in order to generate new iterates. SQP-type algorithms, however, are often at an ad... |

74 | An Interior Point Algorithm for Large Scale Nonlinear Programming, Optimization
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Citation Context ...tilting parameter starts out positive and asymptotically approaches zero. Recently there has been a great deal of interest in interior point algorithms for nonconvex nonlinear programming (see, e.g., =-=[5, 6, 24, 4, 16, 23]-=-). Such algorithms generate feasible iterates and typically only require the solution of linear systems of equations in order to generate new iterates. SQP-type algorithms, however, are often at an ad... |

59 |
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Citation Context ...tilting parameter starts out positive and asymptotically approaches zero. There has been a great deal of interest recently in interior point algorithms for nonconvex nonlinear programming (see, e.g., =-=[5, 6, 26, 4, 18, 25]-=-). Such algorithms generate feasible iterates and typically require only the solution of linear systems of equations in order to generate new iterates. SQP-type algorithms, however, are often at an ad... |

55 | Tits. User’s guide for cfsqp version 2.5: A C code for solving (large scale) constrained nonlinear (minimax) optimization problems, generating iterates satisfying all inequality constraints
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Citation Context ...sen. The algorithm developed by Panier and Tits in [15], and analyzed under weaker assumptions by Qi and Wei in [20], has enjoyed a great deal of success in practice as implemented in the FFSQP/CFSQP =-=[26, 12]-=- software packages. We will refer to their algorithm throughout this paper as FSQP. 3 In [15], instead of directly perturbing QP 0 (x; H), tilting is accomplished by replacing d 0 with the convex comb... |

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42 |
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Citation Context ... j 0 = 1 and # j 0 = #C j 0 (= #), j = 1, . . . , m n . All experiments were run on a Sun Microsystems Ultra 5 workstation. For the first set of numerical tests, we selected a number of problems from =-=[9]-=- which provided feasible initial points and contained no equality constraints. The results are reported in Table 1, where the performance of our implementation of RFSQP is compared with that of CFSQP ... |

35 |
Perturbed Kuhn-Tucker points and rates of convergence for a class of nonlinear programming algorithms
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Citation Context ..., there is no loss of generality is assuming that H k k2K 0 \Gamma! H for some symmetric positive definite H . In view of Lemmas 10 and 11, we may thus invoke a result due to Robinson (Theorem 2.1 in =-=[21]-=-) to conclude that, in view of Lemma 2(ii), (d k ; fl k ) k2K 0 \Gamma! (0; 0);sk k2K 0 \Gamma!s;sk k2K 0 \Gamma!s; a contradiction. Hence the first two claims hold, as does (12). Next, proceeding aga... |

34 |
On the constant positive linear dependence condition and its application to SQP methods
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Citation Context ...ergence. Such a hybrid scheme could give slow convergence if a poor initial point is chosen. The algorithm developed by Panier and Tits in [15], and analyzed under weaker assumptions by Qi and Wei in =-=[20]-=-, has enjoyed a great deal of success in practice as implemented in the FFSQP/CFSQP [26, 12] software packages. We will refer to their algorithm throughout this paper as FSQP. 3 In [15], instead of di... |

29 | On combining feasibility, descent and superlinear convergence in inequality constrained optimization
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- 1993
(Show Context)
Citation Context ...t x is a KKT point for (P ). 3.2 Local Convergence While the details are often quite different, overall the analysis in this section is inspired by and occasionally follows that of Panier and Tits in =-=[14, 15]-=-. The key result is Proposition 1 which states that, under appropriate assumptions, the arc search eventually accepts the full step of one. With this and the fact, to be established along the way, tha... |

29 | A primal-dual interiorpoint method for nonlinear programming with strong global and local convergence properties
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Citation Context ...tilting parameter starts out positive and asymptotically approaches zero. Recently there has been a great deal of interest in interior point algorithms for nonconvex nonlinear programming (see, e.g., =-=[5, 6, 24, 4, 16, 23]-=-). Such algorithms generate feasible iterates and typically only require the solution of linear systems of equations in order to generate new iterates. SQP-type algorithms, however, are often at an ad... |

29 |
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(Show Context)
Citation Context ...ble set. A number of approaches have been considered in the literature for generating feasible directions and, specifically, tilting the SQP direction. Early feasible direction algorithms (see, e.g., =-=[27, 17]-=-) were first-order methods, i.e. only first derivatives were used and no attempt was made to accumulate and use second-order information. Furthermore, search directions were often computed via linear ... |

18 | COPS: Large-scale nonlinearly constrained optimization problems
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(Show Context)
Citation Context ...me expended in linear algebra, specifically in solving the QP and linear least squares subproblems, should be much less. To assess this, we carried out comparative tests on the COPS suite of problems =-=[3]-=-. The first five problems from the COPS set [3] were considered, as these problems either do not involve nonlinear equality constraints or are readily reformulated without such constraints. (Specifica... |

17 |
A QP-free, globally convergent, locally super-linearly convergent algorithm for inequality constrained optimization
- Panier, Tits, et al.
(Show Context)
Citation Context |

17 | Tits, “An SQP algorithm for finely discretized continuous minimax problems and other minimax problems with many objective functions
- Zhou, L
(Show Context)
Citation Context ...important for engineering design is the incorporation of a scheme to efficiently handle very large sets of constraints and/or objectives. We will examine schemes along the lines of those developed in =-=[11, 25]-=-. Further, work remains to be done to exploit the close relation35 ship between the two least squares problems and the quadratic program. A careful implementation should be able to use these relations... |

15 |
Tits. A superlinearly convergence feasible method for the solution of inequality constrained optimization problems
- Panier, L
- 1987
(Show Context)
Citation Context ...for solving (P ). For an excellent recent survey of SQP algorithms, and the theory behind them, see [2]. Denote the feasible set for (P ) by X \Delta = f x 2 R n j g j (x)s0; j = 1; : : : ; m g: 1 In =-=[17, 8, 14, 15, 1], variatio-=-ns on the standard SQP iteration for solving (P ) are proposed which generate iterates lying within X. Such methods are sometimes referred to as "Feasible SQP" (or FSQP) algorithms. It was o... |

13 | User’s guide for QPOPT 1.0: a Fortran package for quadratic programming
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- 1995
(Show Context)
Citation Context ...r the resultant sequence {H k } will, in fact, satisfy Assumption 6, this update scheme is known to perform very well in practice. All QPs and linear least squares subproblems were solved using QPOPT =-=[7]-=-. For comparison's sake, QPOPT was also used to solve the QP subproblems in CFSQP. While the default QP solver for CFSQP is the public domain code QLD (see [24]), we opted for QPOPT because it allows ... |

13 | Nonlinear equality constraints in feasible sequential quadratic programming - Lawrence, Tits - 1994 |

6 |
QLD: A Fortran Code for Quadratic Programming, User's Guide
- Schittkowski
- 1986
(Show Context)
Citation Context ...quares subproblems were solved using QPOPT [7]. For comparison sake, QPOPT was also used to solve the QP subproblems in CFSQP. While the default QP solver for CFSQP is the public domain code QLD (see =-=[22]), we opte-=-d for QPOPT because it allows "warm starts" and thus is fairer to CFSQP in the comparison with the implementation of FSQP 0 (since more QPs are solved with the former). For alls QPs in both ... |

5 |
Tits. Feasible sequential quadratic programming for finely discretized problems from SIP
- Lawrence, L
- 1998
(Show Context)
Citation Context ...important for engineering design is the incorporation of a scheme to efficiently handle very large sets of constraints and/or objectives. We will examine schemes along the lines of those developed in =-=[11, 25]-=-. Further, work remains to be done to exploit the close relation35 ship between the two least squares problems and the quadratic program. A careful implementation should be able to use these relations... |

4 | A variant of the Topkis-Veinott method for solving inequality constrained optimization problems
- Birge, Qi, et al.
(Show Context)
Citation Context ...ilable for solving (P). For an excellent recent survey of SQP algorithms, and the theory behind them, see [2]. Denote the feasible set for (P) by X # = { x # R n | g j (x) # 0, j = 1, . . . , m }. In =-=[19, 8, 16, 17, 1], variatio-=-ns on the standard SQP iteration for solving (P) are proposed which generate iterates lying within X. Such methods are sometimes referred to as "feasible SQP" (FSQP) algorithms. It was obser... |

3 |
Constant positive linear independence, KKT points and convergence of feasible SQP methods
- Qi, Wei
- 1997
(Show Context)
Citation Context ...ergence. Such a hybrid scheme could give slow convergence if a poor initial point is chosen. The algorithm developed by Panier and Tits in [12], and analyzed under weaker assumptions by Qi and Wei in =-=[17]-=-, has enjoyed a great deal of success in practice as implemented in the FFSQP/CFSQP [23, 10] software packages. We will refer to their algorithm throughout this paper as FSQP. In [12], instead of dire... |

2 |
A successive quadratic programming based feasible directions algorithm
- Herskovits, Carvalho
- 1986
(Show Context)
Citation Context ...ilable for solving (P). For an excellent recent survey of SQP algorithms, and the theory behind them, see [2]. Denote the feasible set for (P) by X # = { x # R n | g j (x) # 0, j = 1, . . . , m }. In =-=[19, 8, 16, 17, 1], variatio-=-ns on the standard SQP iteration for solving (P) are proposed which generate iterates lying within X. Such methods are sometimes referred to as "feasible SQP" (FSQP) algorithms. It was obser... |

1 |
Exact Penalty Functions for Finite Dimensional and Control Optimization Problems
- Maratos
- 1978
(Show Context)
Citation Context ...of H is often assumes as it ensures existence and uniqueness of such solution. With an appropriate merit function, line search procedure, Hessian approximation rule, and (if necessary) Maratos effect =-=[13]-=- 2 avoidance scheme, the SQP iteration is known to be globally and locally superlinearly convergent (see, e.g., [2]). A feasible direction at a point x 2 X is defined as any vector d in R n such that ... |

1 |
User's Guide for FSQP Version 3.7: A FORTRAN Code for Solving Nonlinear (Minimax
- Zhou, Tits, et al.
- 1997
(Show Context)
Citation Context ...sen. The algorithm developed by Panier and Tits in [15], and analyzed under weaker assumptions by Qi and Wei in [20], has enjoyed a great deal of success in practice as implemented in the FFSQP/CFSQP =-=[26, 12]-=- software packages. We will refer to their algorithm throughout this paper as FSQP. 3 In [15], instead of directly perturbing QP 0 (x; H), tilting is accomplished by replacing d 0 with the convex comb... |