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1496 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 Improved Feasible SQP Algorithm for Nonlinear Programs with Equality Constrained Sub- Problems Zhijun Luo 1 , Guohua Chen 3 and Simei Luo 4 Department of Mathematics & Applied Mathematics, Hunan University of Humanities, Science and Technology, Loudi, China Email: ldlzj123@163.com Zhibin Zhu 2 School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, China Email: zhu_zhibin@163.com Abstract—This paper proposed an improved feasible sequential quadratic programming (FSQP) method for nonlinear programs. As compared with the existing SQP methods which required solving the QP sub-problem with inequality constraints in single iteration, in order to obtain the feasible direction, the method of this paper is only necessary to solve an equality constrained quadratic programming sub-problems. Combined the generalized projection technique, a height-order correction direction is yielded by explicit formulas, which can avoids Maratos effect. Furthermore, under some mild assumptions, the algorithm is globally convergent and its rate of convergence is one-step superlinearly. Numerical results reported show that the algorithm in this paper is effective. Index Terms—Nonlinear programs, FSQP method, Equality constrained quadratic programming, Global convergence, Superlinear convergence rate I. INTRODUCTION Consider the following nonlinear programs min f( x) s.. t g ( x) ≤0, j∈ I = {1,2, , m}, j Where f ( x), g ( ): n j x R → R( j∈I) are continuously differentiable functions. Denote the feasible set for (1) by n X = { x∈R | g j ( x) ≤0, j∈ I} . The Lagrangian function associated with (1) is defined as follows: Lx ( , λ) f( x) λ g( x) m = +∑ j= 1 A point x ∈ X is said to be a KKT point of (1), if it is satisfies the equalities This work was supported in part by the National Natural Science Foundation (11061011) of China, and the Educational Reform Research Fund of Hunan University of Humanities, Science and Technology (NO.RKJGY1030), corresponding author, E-mail: ldlzj123@163.com . j j (1) m ∇ f( x) + λ ∇ g ( x) = 0, j= 1 λ g ( x) = 0, j∈I, j j ∑ where λ = ( λ1 , , λ ) T m is nonnegative, and λ is said to be the corresponding KKT multiplier vector. Method of Sequential Quadratic Programming (SQP) is an important method for solving nonlinearly constrained optimization [1, 2, 18]. It generates iteratively the main search direction d 0 by solving the following quadratic programming (QP) sub-problem: j T 1 T min ∇ f( x) d + d Hd 2 T s.. t g ( x) + ∇g ( x) d ≤0, j∈I, j n n where H ∈ R × is a symmetric positive definite matrix. However, such type SQP algorithms have two serious shortcomings: 1) SQP algorithms require that the relate QP subproblems (2) must be consistency; 2) There exists Matatos effect. Many efforts have been made to overcome the shortcomings through modifying the quadratic subproblem (2) and the direction d [4, 5, 7, 8]. Some algorithms solve the problem (1) by using the idea of filter method or trust-region [13, 16, 17]. For the problem (2), it is also a hot topic to solve the QP problem like (2) in the field of optimization. By using the idea of active constraints set, some algorithms solve step by step a series of corresponding QP problems with only equality constraints to obtain the optimum solution to the QP sub-problem (2). P. Spellucci [6] proposed a new method, the d 0 is obtained by solving QP subproblem with only equality constraints: j j (2) © 2013 ACADEMY PUBLISHER doi:10.4304/jcp.8.6.1496-1503
JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1497 T 1 T min ∇ f( x) d + d Hd 2 T s.. t g ( x) +∇ g ( x) d = 0, j∈A⊆ I, j j where the so-called working set A ⊆ I is suitably determined. If d 0 = 0 and λ ≥ 0 ( λ is said to be the corresponding KKT multiplier vector.), the algorithm stops. The most advantage of these algorithms is merely necessary to solve QP sub-problems with only equality constraints. However, if d 0 = 0 , but λ < 0 , the algorithm will not implement successfully. In [10], proposed an SQP method for general constrained optimization. Firstly, make use of the technique which handle the general constrained optimization as an inequality parametric programming, then, consider a new quadratic programming with only equality constraints as follow: T 1 T min ∇ f( x) d + d Hd 2 T s.. t g ( x) +∇ g ( x) d =−min{0, π ( x)}, j∈J( x). j j Where π ( x) is a suitable vector, J ( x ) is a suitable approximate active set. But the QP problems may no solution under some conditions. Recently, Zhu [14] Consider the following QP sub-problem: T 1 T min ∇ f( x) d + d Hd 2 T s.. t p ( x) +∇ g ( x) d = 0, j∈L. j j where p ( x ) is a suitable vector, L is a suitable j approximate active, which guarantees to hold that if d 0 = 0 , then x is a KKT point of (1), i.e., if d 0 = 0 , then it holds that λ ≥ 0 . Depended strictly on the strict complementarity, which is rather strong and difficult for testing, the superlinear convergence properties of the SQP algorithm are obtained. For avoiding the superlinear convergence depend strictly on the strict complementarity, Another some SQP algorithms (see [15]) have been proposed, however it is regretful that these algorithms are infeasible SQP type and nonmonotone. In [16], a feasible SQP algorithm is proposed. Using generalized projection technique, the superlinear convergence properties are still obtained under weaker conditions without the strict complementarity. We will develop an improved feasible SQP method for solving optimization problems based on the one in [14]. The traditional FSQP algorithms, in order to prevent iterates from leaving the feasible set, and avoid Maratos effect, it needs to solve two or three QP sub-problems like (2). In our algorithm, per single iteration, it is only necessary to solve an equality constrained quadratic programming, which is very similar to (4). Obviously, it is simpler to solve the equality constrained QP problem than to solve the QP problem with inequality constraints. In order to void the Maratos effect, combined the generalized projection technique, a height-order correction direction is computed by an explicit formula, and it plays an important role in avoiding the strict (3) (4) complementarity. Furthermore, its global and superlinear convergence rate is obtained under some suitable conditions. This paper is organized as follows: In Section II, we state the algorithm; the well-defined of our approach is also discussed, the accountability of which allows us to present global convergence guarantees under common conditions in Section III, while in Section IV we deal with superlinear convergence. Finally, in Section V, numerical experiments are implemented. II. DESCRIPTION OF ALGORITHM The active constraints set of (1) is denoted as follows: I( x) = { j∈ I | g ( x) = 0, j∈ I}. (5) Now, the following algorithm is proposed for solving the problem (1). Algorithm A: Step 0 Initialization: Given a starting point 0 x ∈ X , and an initial n n symmetric positive definite matrix H 0 ∈ R × . Choose 1 parameters ε0 ∈(0,1), α∈(0, ), τ ∈ (2,3) . Set k = 0 ; 2 Step 1. Computation of an approximate active set J k . Step 1.1. For the current point 0 j k x k ε ( x ) = ε ∈ (0,1). i ∈ X , set i = 0, Step 1.2. If det( A ( x k ) T A( x k )) ≥ ε ( x k ) , let i i i k k k Jk = J( x ), Ak = A( x ), i( x ) = i , and go to Step 2. Otherwise go to Step 1.3, where k k k J ( x ) = { j∈I | −ε ( x ) ≤ g ( x ) ≤0}, i i j k k k A( x ) = ( ∇g ( x ), j∈J ( x )). i i i k 1 k Step 1.3. Let i = i+ 1, εi( x ) = εi − 1( x ) , and go to 2 Step1. 2. k Step 2. Computation of the vector d 0 . Step 2.1 B A A A v v j J B f x T −1 T k k k k = ( k k) k , = ( j, ∈ k) = − k∇ ( ), k k ⎧ , 0 k ⎪ − vj vj < k k pj = ⎨ p = ( pj, j∈Jk). k k ⎪⎩ g j( x ), vj ≥ 0 Step 2.2 Solve the following equality constrained QP k Sub problem at x : k T 1 T min ∇ f( x ) d + d Hkd 2 k k T s. t. p +∇ g ( x ) d = 0, j∈J . j j k k Let d0 be the KKT point of (8), and k k b = ( b , j∈J ) be the corresponding multiplier vector. j k k If d 0 = 0 , STOP. Otherwise, CONTINUE; (6) (7) (8) © 2013 ACADEMY PUBLISHER
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Journal of Computers ISSN 1796-203X
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Corn Moisture Measurement using a C
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Call for Papers and Special Issues
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(Contents Continued from Back Cover
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