# On the Use of Biased-Randomized Algorithms for Solving Non-Smooth Optimization Problems
## 1 Introduction  
Optimization models are used in many practical situations to represent decision-making challenges in areas such as computational finance, transportation and logistics, telecommunication networks, smart cities, etc. [1]. Many of these challenges can be transformed into optimization problems (OPs) that can be then solved using a plethora of methods of both exact and approximatenature. Typically, solving an OP implies exploring a vast solution space while searching for one solution that minimizes or maximizes a given objective function. In addition, the solution has to satisfy a series of constraints in order to be a feasible one [2]. It is frequent to model these OPs by using linear programming (LP), integer programming (IP), or mixed integer linear programming (MILP) methods. Unfortunately, in many real-life situations, these OPs are also _NP-hard_, which implies that the computing time requested to find an optimal solution grows extraordinarily fast as the size of the problem increases [3]. Hence, one has to make use of heuristic-based algorithms if reasonably good solutions are needed in short computing times for large-scale _NP-hard_ OPs [4]. Moreover, the effective use of exact methods might be also limited whenever the mathematical model does not comply with desirable properties such as convexity or smoothness. In particular, the existence of non-convex and non-smooth objective functions might limit the efficiency of gradient-based optimization methods. [PERSON] and [PERSON] [5] discuss a non-smooth OP called the minimum sum-of-squares clustering problem. According to the authors, previously employed approaches such as dynamic programming, branch-and-bound, or the \(k\)-means algorithm are efficient only for small instances of this problem. The authors also support the idea that the use of heuristic-based approaches becomes necessary for large-size instances. Similarly, [PERSON] et al. [6] analyze another non-smooth OP related to the facility location problem in a wireless sensor network. [PERSON] et al. [7] study non-smooth power-flow problems, while [PERSON] et al. [8] propose an adaptive hybrid differential evolution algorithm to cope with a non-smooth version of the dynamic economic dispatch problem. To the best of our knowledge, however, there is a lack of publications considering realistic non-smooth cost functions in many OPs. Nevertheless, OPs with soft constraints might frequently appear in real-life applications. As discussed in [PERSON] et al. [9], \"in real-world simulations, time windows and capacity constraints can be often violated to some extent\". Hence, for example, in cost minimization, problems violating these soft constraints might generate penalty costs that might be taken into account in the objective function. These penalty costs will typically come in the form of a piecewise cost function, which can transform the objective function into a non-smooth one. This paper reviews different examples of OPs with non-smooth objective functions and then analyzes how biased-randomized algorithms (BRAs) can constitute an effective methodology to generate reasonably good solutions in very short computing times. As described in [PERSON] et al. [10], BRAs make use of skewed probability distributions to integrate a \"biased\" (nonuniform) random behavior into a heuristic. This allows one to quickly generate a large set of alternative good solutions by simply changing the seed of the pseudo-random number generator [11; 12]. Hence, each execution of the BRA can be seen as an individual \"agent\" searching the solution space following the logic behind the heuristic but starting from a different point and using a different searching (Figure 1). Moreover, the execution of these BRA agents can be performed in parallel, thus consuming virtually the same time as the original heuristic (i.e., milliseconds in most cases).