Cyclic schedules for r irregularly occurring event

Consider r irregular polygons with vertices on some circle. Authors explains how the polygons should be arranged to minimize some criterion function depending on the distances between adjacent vertices. A solution of this problem is given. It is based on a decomposition of the set of all schedules into local regions in which the optimization problem is convex. For the criterion functions minimize the maximum distance and maximize the minimum distance the local optimization problems are related to network flow problems which can be solved efficiently. If the sum of squared distances is to be minimized a locally optimal solution can be found by solving a system of linear equations. For fixed r the global problem is polynomially solvable for all the above-mentioned objective functions. In the general case, however, the global problem is NP-hard

Fuzzy set applications in engineering optimization: Multilevel fuzzy optimization

A formulation for multilevel optimization with fuzzy objective functions is presented. With few exceptions, formulations for fuzzy optimization have dealt with a one-level problem in which the objective is the membership function of a fuzzy set formed by the fuzzy intersection of other sets. In the problem examined here, the goal set G is defined in a more general way, using an aggregation operator H that allows arbitrary combinations of set operations (union, intersection, addition) on the individual sets Gi. This is a straightforward extension of the standard form, but one that makes possible the modeling of interesting evaluation strategies. A second, more important departure from the standard form will be the construction of a multilevel problem analogous to the design decomposition problem in optimization. This arrangement facilitates the simulation of a system design process in which different components of the system are designed by different teams, and different levels of design detail become relevant at different time stages in the process: global design features early, local features later in the process

Decomposition Approaches for Building Design Optimization

Building performance simulation can be integrated with optimization to achieve high-performance building design objectives such as low carbon emission and cost-effectiveness by holistically considering design variables across different disciplines. However, the complexity of the design problem increases greatly with increasing dimensionality. In some cases, solving high-dimension problems is not technically feasible nor time-efficient. Decomposition is one way to reduce the complexity and dimensionality of optimization problems. However, the decomposed optimization might achieve local optimum. Therefore, deploying appropriate decomposition strategies to achieve global optimum is paramount. This study investigates the deployment of hierarchical and parallel decomposition for building design optimization problems to ensure identification of global optimum. The feasibility of combining sensitivity analysis and decomposition is also explored. At the end of this study, some recommendations are given to help select an appropriate approach in practice. First, this thesis proposes a hierarchical decomposition. Hierarchical decomposition divides an optimization problem into several interconnected subproblems solved sequentially. The proposed approach is applied to the multi-objective optimization problem that minimizes buildings' operating costs and carbon emissions. The results show that the hierarchical decomposition approach can reduce the number of simulations while achieving global optimums. Second, this thesis proposes a parallel decomposition. Parallel decomposition divides the original problem into several smaller subproblems to be solved separately, and potentially, concurrently. The proposed parallel decomposition approach is applied to solve the single-objective optimization problems of a benchmark function and a low-rise office building. The results show that the proposed approach finds the global optimum and takes less computation time than optimization without decomposition. Third, this thesis explores the feasibility of combining sensitivity analysis with decomposition for dimensionality reduction. The efficiency and accuracy of different methods are compared through three case studies. The proposed hierarchical and parallel decomposition approaches can be applied individually or combined into a hybrid decomposition approach. This thesis concludes with some recommendations to help choose a decomposition approach to solve building design optimization problems

Multidisciplinary Design Optimization with Mixed Integer Quasiseparable Subsystems

Numerous hierarchical and nonhierarchical decomposition strategies for the optimization of large scale systems, comprised of interacting subsystems, have been proposed. With a few exceptions, all of these strategies have proven theoretically unsound. Recent work considered a class of optimization problems, called quasiseparable, narrow enough for a rigorous decomposition theory, yet general enough to encompass many large scale engineering design problems. The subsystems for these problems involve local design variables and global system variables, but no variables from other subsystems. The objective function is a sum of a global system criterion and the subsystems' criteria. The essential idea is to give each subsystem a budget and global system variable values, and then ask the subsystems to independently maximize their constraint margins. Using these constraint margins, a system optimization then adjusts the values of the system variables and subsystem budgets. The subsystem margin problems are totally independent, always feasible, and could even be done asynchronously in a parallel computing context. An important detail is that the subsystem tasks, in practice, would be to construct response surface approximations to the constraint margin functions, and the system level optimization would use these margin surrogate functions. The present paper extends the quasiseparable necessary conditions for continuous variables to include discrete subsystem variables, although the continuous necessary and sufficient conditions do not extend to include integer variables

Multi agent collaborative search based on Tchebycheff decomposition

This paper presents a novel formulation of Multi Agent Collaborative Search, for multi-objective optimization, based on Tchebycheff decomposition. A population of agents combines heuristics that aim at exploring the search space both globally (social moves) and in a neighborhood of each agent (individualistic moves). In this novel formulation the selection process is based on a combination of Tchebycheff scalarization and Pareto dominance. Furthermore, while in the previous implementation, social actions were applied to the whole population of agents and individualistic actions only to an elite sub-population, in this novel formulation this mechanism is inverted. The novel agent-based algorithm is tested at first on a standard benchmark of difficult problems and then on two specific problems in space trajectory design. Its performance is compared against a number of state-of-the-art multi objective optimization algorithms. The results demonstrate that this novel agent-based search has better performance with respect to its predecessor in a number of cases and converges better than the other state-of-the-art algorithms with a better spreading of the solutions