A new superlinearly convergent algorithm of combining QP subproblem with system of linear equations for nonlinear optimization

Abstract

In this paper, a class of optimization problems with nonlinear inequality constraints is discussed. Based on the ideas of sequential quadratic programming algorithm and the method of strongly sub-feasible directions, a new superlinearly convergent algorithm is proposed. The initial iteration point can be chosen arbitrarily for the algorithm. At each iteration, the new algorithm solves one quadratic programming subproblem which is always feasible, and one or two systems of linear equations with a common coefficient matrix.Moreover, the coefficient matrix is uniformly nonsingular. After finite iterations, the iteration points can always enter into the feasible set of the problem, and the search direction is obtained by solving one quadratic pro- algorithm possesses global and superlinear convergence under some suitable assumptions without the strict complementarity. Finally, some preliminary numerical experiments are reported to show that the algorithm is promising.

A superlinearly convergent hybrid algorithm for solving nonlinear programming

“…Subsequently, Bonnans, et.al., propose a new feasible SQP algorithm, by modifying this convex combination technique, for solving problem (NIO) in [2]. Recently, Jian, et.al., present a strongly sub-feasible SQP algorithm for solving problem (NIO) [10] with the help of this convex combination formula.…”

“…from (9), (10) and (12). Then, combining the constraints of subproblem (5) with (8), it follows that…”

“…For i ∈ I(x * ), since lim k→+∞ f i (x k ) = f i (x * ) = 0 and lim k→+∞ Υ(x k ) = 0 it holds that from (10)…”

“…Then, for further showing the performance of our algorithm, we compare the number of iterations required by ALGO 1 under Case II to those required by Wang-Chen-He's algorithm [23] (shorted by ALGO-WCH) and algorithm in [10] (denoted by ALGO 2.1), and the corresponding results are shown in the following Figure 3. The optimality tolerance is the same as in [23] and [10], respectively. The left figure of Figure 3 gives the performance of ALGO 1 under Case II and ALGO-WCH, the test problems are the same as in [23].…”

“…The left figure of Figure 3 gives the performance of ALGO 1 under Case II and ALGO-WCH, the test problems are the same as in [23]. For some large problems in Table 4 in [10], the corresponding performance of ALGO 1 under Case II compared to ALGO 2.1 are shown in the right figure of Figure 3. From the results in Figure 3, it is quite clear that the performance of ALGO 1-II is better than that of ALGO-WCH, ALGO 2.1, respectively, i.e., ALGO 1-II has the most wins compare with ALGO-WCH and ALGO 2.1.…”