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Abstract: In this paper, by means of “working set” technique for determining the active set and the idea of primal–dual interior-point method, a new feasible QP-free algorithm for solving inequality constrained optimization problems is presented. At each iteration, the algorithm solves only three reduced systems of linear equations with common coefficient matrix. Moreover, the initial iteration point can be at constraint boundary and the coefficient matrix is uniformly nonsingular without the strict complementarity. We also prove that the proposed algorithm obtains global and superlinear convergence. At last, preliminary numerical results are reported. [Copyright &y& Elsevier]

Abstract: This paper concerns a nonmonotone line search technique and its application to the trust region method for unconstrained optimization problems. In our line search technique, the current nonmonotone term is a convex combination of the previous nonmonotone term and the current objective function value, instead of an average of the successive objective function values that was introduced by Zhang and Hager [H. Zhang, W.W. Hager, A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim. 14 (4) (2004) 1043–1056]. We incorporate this nonmonotone scheme into the traditional trust region method such that the new algorithm possesses nonmonotonicity. Unlike the traditional trust region method, our algorithm performs a nonmonotone line search to find a new iteration point if a trial step is not accepted, instead of resolving the subproblem. Under mild conditions, we prove that the algorithm is global and superlinear convergence holds. Primary numerical results are reported. [Copyright &y& Elsevier]

Abstract: In this paper, a kind of nonlinear optimization problems with nonlinear inequality constraints are discussed, and a new SQP feasible descent algorithm for solving the problems is presented. At each iteration of the new algorithm, a convex quadratic program (QP) which always has feasible solution is solved and a master direction is obtained, then, an improved (feasible descent) direction is yielded by updating the master direction with an explicit formula, and in order to avoid the Maratos effect, a height-order correction direction is computed by another explicit formula of the master direction and the improved direction. The new algorithm is proved to be globally convergent and superlinearly convergent under mild conditions without the strict complementarity. Furthermore, the quadratic convergence rate of the algorithm is obtained when the twice derivatives of the objective function and constrained functions are adopted. Finally, some numerical tests are reported. [Copyright &y& Elsevier]

In this paper, we combine the new trust region subproblem proposed in [1] with the nonmonotone technique to propose a new algorithm for unconstrained optimization—the nonmonotone adaptive trust region method. The local and global convergence properties of the nonmonotone adaptive trust region method are proved. Its efficiency is tested by numerical results. [Copyright &y& Elsevier]