# Large-Scale Truss-Sizing Optimization with Enhanced Hybrid HS Algorithm
## 1 Introduction  
Metaheuristic optimization methods inspired by evolution theory, life sciences and zoology, physics and astronomy, human sciences, etc., are successfully utilized in science and engineering. For example, genetic algorithms (GA) [1], differential evolution [2], simulated annealing (SA) [3], particle swarm optimization (PSO) [4], ant colony optimization (ACO) [5], firefly algorithm (FFA) [6], cuckoo search (CS) [7], ant lion optimizer [8], Tabu search (TS) [9], harmony search (HS) [10], teaching-learning based optimization (TLBO) [11], JAYA [12], big bang-big crunch (BBBC) [13], charged system search (CSS) [14], ray optimization (RO) [15], colliding bodies optimization [16], water evaporation optimization (WEO) [17], cyclical parthenogenesis algorithm (CPA) [18], and coyote optimization algorithm (COA) [19] are representative metaheuristic methods with several variants documented in the optimization literature. Generally speaking, optimization algorithms may be trajectory-based or population-based. In the former case, a single design is elaborated in each iteration, thus building a \"trajectory\" connecting the initial solution with the optimal solution and passing throughall intermediate solutions found in each iteration. Trajectory-based algorithms include gradient-based methods using linear or quadratic approximations of the optimization problem as well as some metaheuristic algorithms such as, for example, simulated annealing, randomized local search and simplified variants of differential evolution. Population-based algorithms update a population of candidate solutions until the search process converges to the optimum design after a predefined number of iterations or function evaluations. This category practically includes all metaheuristic algorithms except those classified above as trajectory-based algorithms. While metaheuristic algorithms practically became the standard approach to engineering optimization (see, for example, the review articles and books [20; 21; 22; 23; 24] focusing on structural optimization), the \"no free lunch\" theorem [25; 26] had an important consequence in the fact that no metaheuristic algorithm can prove itself superior over all other algorithms in all problems. A general-purpose universal optimization algorithm does not exist from the theoretical point of view. However, an algorithm can outperform its competitors if it is specialized to the specific problem at hand. For this reason, most of the metaheuristic algorithms lost their appeal just after a very few years. This is not the case of Harmony Search (HS) [10], which was developed almost 20 years ago, but remains a very popular optimization algorithm. The HS method is a population-based metaheuristic algorithm that simulates the process of searching for a perfect state of harmony performed by jazz players. The relationship of HS to music is well summarized in [PERSON] [27]. Three possible options are available to a skilled musician who is improvising: (1) play any famous piece of music (a series of pitches in harmony) exactly from his or her memory; (2) play something similar to a known piece (thus adjusting the pitch slightly); (3) compose new or random notes. These three options were formalized by [PERSON] et al. [10] into the harmony search optimization algorithm. A set of \(N_{\text{POP}}\) randomly generated solutions are stored in a matrix called harmony memory [HM]. New trial designs may be extracted from [HM] or randomly selected according to the value taken by the harmony memory considering rate (_HMCR_) parameter. Design variables may be modified according to the value taken by the pitch adjustment rate (_PAR_) parameter; the bandwidth parameter (_bw_) quantifies the amount of variation given to a variable in the pitch adjustment operation. Structural optimization is an important field of engineering concerned with the optimum design of structures [28; 29; 30].