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The Chinese postman problem is a famous and classical problem in graph theory. This paper introduces a new variant of this problem, namely, the time-dependent rural postman problem, which is motivated by applications, involving scheduling with time-dependent processing time. We present an arc-path formulation for the problem with a constraint set that is divided into two parts. The first part is linear and has a strong combinatorial structure, which defines the polytope of the arc-path alternation sequence (APAS), while the second part is nonlinear and is closely related to time-dependent travel time. First, we investigate the facial structure of the APAS polytope and present several facet inequalities. Second, considering the travel and service time functions as piecewise functions, we linearize the nonlinear inequalities and propose several strong, valid inequalities. Finally, we propose a cutting plane algorithm and report numerical results on several randomly generated instances. [ABSTRACT FROM PUBLISHER]

This paper first proposes a trust region algorithm to obtain a stationary point of unconstrained multiobjective optimisation problem. Under suitable assumptions, the global convergence of the new algorithm is established. We then extend the trust region method to solve the non-smooth multiobjective optimisation problem. [ABSTRACT FROM PUBLISHER]

Detection of copositivity plays an important role in combinatorial and quadratic optimization. Recently, an algorithm for copositivity detection by simplicial partition has been proposed. In this paper, we develop an improved depth-first simplicial partition algorithm which reduces memory requirements significantly and therefore enables copositivity checks of much larger matrices - of size up to a few thousands instead of a few hundreds. The algorithm has been investigated experimentally on a number of MaxClique problems as well as on generated random problems. We present numerical results showing that the algorithm is much faster than a recently published linear algebraic algorithm for copositivity detection based on traditional ideas - checking properties of principal sub-matrices. We also show that the algorithm works very well for solving MaxClique problems through copositivity checks. [ABSTRACT FROM AUTHOR]