In this paper, we investigate and develop a new filled function method for solving integer programming problems with constraints. By adopting the appropriate equivalent transformation method, these problems are transformed into a class of box-constrained integer programming problems. Then, an effective nonparametric filled function is constructed, and a new global optimization algorithm is designed using the discrete steepest descent method. Numerical experiments illustrate that this algorithm has effectiveness, feasibility, and better global optimization ability.

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Quadratically constrained quadratic programs (QCQP), which often appear in engineering practice and management science, and other fields, are investigated in this paper. By introducing appropriate auxiliary variables, QCQP can be transformed into its equivalent problem (EP) with non-linear equality constraints. After these equality constraints are relaxed, a series of linear relaxation subproblems with auxiliary variables and bound constraints are generated, which can determine the effective lower bound of the global optimal value of QCQP. To enhance the compactness of sub-rectangles and improve the ability to remove sub-rectangles, two rectangle-reduction strategies are employed. Besides, two ϵ-subproblem deletion rules are introduced to improve the convergence speed of the algorithm. Therefore, a relaxation and bound algorithm based on auxiliary variables are proposed to solve QCQP. Numerical experiments show that this algorithm is effective and feasible.