Abstract In deterministic global optimization, techniques for linear relaxation of a non-convex program are used in the lower bound calculation phase. To achieve this phase, most deterministic global optimization codes use reformulation-linearization techniques. However, there exist also two interval-based polyhedral relaxation techniques which produce reliable bounds without adding new auxiliary variables, and which can take into account mathematical operations and most transcendental functions: (i) the affine relaxation technique, used in the IBBA code, based on affine forms and affine arithmetic, and (ii) the extremal Taylor technique, used in the Ibex-Opt code, which is based on a specific interval-based Taylor form. In this paper, we describe how these two interval-based linear relaxation techniques can be hybridized. These two approaches appear to be complementary, and such a hybrid method performs well on a representative sample of constrained global optimization instances.

Abstract This study focuses on exhaustive global optimization algorithms over a simplicial feasible set with simplicial partition sets. Bounds on the objective function value and its partial derivative are based on interval automatic differentiation over the interval hull of a simplex. A monotonicity test may be used to decide to either reject a simplicial partition set or to reduce its simplicial dimension to a relative border (at the boundary of the feasible set) facet (or face) by removing one (or more) vertices. A monotonicity test is more complicated for a simplicial sub-set than for a box, because its orientation does not coincide with the components of the gradient. However, one can focus on directional derivatives (DD). In a previous study, we focused on either basic directions, such as centroid to vertex or vertex to vertex directions, or finding the best directional derivative by solving an LP or MIP. The research question of this paper refers to using local search (LS) based sampling of directions from vertex to facet. Results show that most of the monotonic DD found by LP are also found by LS, but with much less computational cost. Notice that finding a monotone direction does not require to find the direction in which a derivative bound is the steepest.

Abstract Branch and Bound (B&B) algorithms in Global Optimization are used to perform an exhaustive search over the feasible area. One choice is to use simplicial partition sets. Obtaining sharp and cheap bounds of the objective function over a simplex is very important in the construction of efficient Global Optimization B&B algorithms. Although enclosing a simplex in a box implies an overestimation, boxes are more natural when dealing with individual coordinate bounds, and bounding ranges with Interval Arithmetic (IA) is computationally cheap. This paper introduces several linear relaxations using gradient information and Affine Arithmetic and experimentally studies their efficiency compared to traditional lower bounds obtained by natural and centered IA forms and their adaption to simplices. A Global Optimization B&B algorithm with monotonicity test over a simplex is used to compare their efficiency over a set of low dimensional test problems with instances that either have a box constrained search region or where the feasible set is a simplex. Numerical results show that it is possible to obtain tight lower bounds over simplicial subsets.

International audience

Interval branch-and-bound (B&B) algorithms are powerful methods which look for guaranteed solutions of global optimisation problems. The computational effort needed to reach this aim, increases exponentially with the problem dimension in the worst case. For separable functions this effort is less, as lower dimensional sub-problems can be solved individually. The question is how to design specific methods for cases where the objective function can be considered separable, but common variables occur in the sub-problems. This paper is devoted to establish the bases of B&B algorithms for separable problems. New B&B rules are presented based on derived properties to compute bounds. A numerical illustration is elaborated with a test-bed of problems mostly generated by combining traditional box constrained global optimisation problems, to show the potential of using the derived theoretical basis.

Spatial Division Multiple Access (SDMA) is a principle of radio resource sharing that separates communication channels in space. It relies on adaptive and dynamic beam-forming technology and well-designed algorithms for resource allocation. As satellite communication systems move towards greater capacity in both the number of users and throughput, SDMA becomes one of the most promising techniques that can achieve these two goals. This paper studies static Frequency Assignment Problem (FAP) in a satellite communication system involving a satellite and a number of users located in a service area. The objective is to maximize the number of users that the system can serve while maintaining the signal to interference plus noise ratio of each user under a predefined threshold. Traditionally, interference is binary and fixed. In this paper, the interference is cumulative and variable depending on how the frequency is assigned. To solve the problem, we work on both discrete and continuous optimizations. Integer linear programming formulations and greedy algorithms are proposed for solving the discrete frequency allocation problem. The solution is further improved by beam moving algorithm which involves continuous adjustment of satellite beams and deals with non-linear change of interference.