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Pancreatic cancer (PC) is an aggressive human cancer. Appropriate methods for the diagnosis and treatment of PC have not been found at the genetic level, thus making epigenetics a promising research path in studies of PC. Histone methylation is one of the most complicated types of epigenetic modifications and has proved crucial in the development of PC. Histone methylation is a reversible process regulated by readers, writers, and erasers. Some writers and erasers can be recognized as potential biomarkers and candidate therapeutic targets in PC because of their unusual expression in PC cells compared with normal pancreatic cells. Based on the impact that writers have on the development of PC, some inhibitors of writers have been developed. However, few inhibitors of erasers have been developed and put to clinical use. Meanwhile, there is not enough research on the reader domains. Therefore, the study of erasers and readers is still a promising area. This review focuses on the regulatory mechanism of histone methylation, and the diagnosis and chemotherapy of PC based on it. The future of epigenetic modification in PC research is also discussed.

In this paper, a superlinearly convergent hybrid algorithm is proposed for solving nonlinear programming. First of all, an improved direction is obtained by a convex combination of the solution of an always feasible quadratic programming (QP) subproblem and a mere feasible direction, which is generated by a reduced system of linear equations (SLE). The coefficient matrix of SLE only consists of the constraints and their gradients corresponding to the working set. Then, the second-order correction direction is obtained by solving another reduced SLE to overcome the Maratos effect and obtain the superlinear convergence. In particular, the convergence rate is proved to be locally one-step superlinear under a condition weaker than the strong second-order sufficient condition and without the strict complementarity. Moreover, the line search in our algorithm can effectively combine the initialization processes with the optimization processes, and the line search conditions are weaker than the previous work. Finally, some numerical results are reported.

The densest k-subgraph (DkS) maximization problem is to find a set of k vertices with maximum total weight of edges in the subgraph induced by this set. This problem is in general NP-hard. In this paper, two relaxation methods for solving the DkS problem are presented. One is doubly nonnegative relaxation, and the other is semidefinite relaxation with tighter relaxation compare with the relaxation of standard semidefinite. The two relaxation problems are equivalent under the suitable conditions. Moreover, the corresponding approximation ratios&rsquo, results are given for these relaxation problems. Finally, some numerical examples are tested to show the comparison of these relaxation problems, and the numerical results show that the doubly nonnegative relaxation is more promising than the semidefinite relaxation for solving some DkS problems.

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 programming subproblem and only one system of linear equations. The new 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.

Multicriterion optimization and Pareto optimality are fundamental tools in economics. In this paper we propose a new relaxation method for solving multiple objective quadratic programming problems. Exploiting the technique of the linear weighted sum method, we reformulate the original multiple objective quadratic programming problems into a single objective one. Since such single objective quadratic programming problem is still nonconvex and NP-hard in general. By using the techniques of lifting and doubly nonnegative relaxation, respectively, this single objective quadratic programming problem is transformed to a computable convex doubly nonnegative programming problem. The optimal solutions of this computable convex problem are (weakly) Pareto optimal solutions of the original problem under some mild conditions. Moreover, the proposed method is tested with two examples and a practical portfolio selection problem. The test problems are solved by \texttt{CVX} package which is a solver for convex optimization. The numerical results show that the proposed method is effective and promising., Comment: 14 pages, 2 figures

In this paper, a class of general nonlinear programming problems with inequality and equality constraints is discussed. Firstly, the original problem is transformed into an associated simpler equivalent problem with only inequality constraints. Then, inspired by the ideals of the sequential quadratic programming (SQP) method and the method of system of linear equations (SLE), a new type of SQP algorithm for solving the original problem is proposed. At each iteration, the search direction is generated by the combination of two directions, which are obtained by solving an always feasible quadratic programming (QP) subproblem and a SLE, respectively. Moreover, in order to overcome the Maratos effect, the higher-order correction direction is obtained by solving another SLE. The two SLEs have the same coefficient matrices, and we only need to solve the one of them after a finite number of iterations. By a new line search technique, the proposed algorithm possesses global and superlinear convergence under some suitable assumptions without the strict complementarity. Finally, some comparative numerical results are reported to show that the proposed algorithm is effective and promising.