Derivatives are an important tool for single-objective optimization. In fact, it is commonly accepted that derivative-based methods present a better performance than derivative-free optimization approaches. In this work, we will show that the same does not apply to multiobjective derivative-based optimization, when the goal is to compute an approximation to the complete Pareto front of a given problem. The competitiveness of Direct MultiSearch (DMS), a robust and efficient derivative-free optimization algorithm, will be stated for derivative-based multiobjective optimization problems. We will then assess the potential enrichment of adding first-order information to the DMS framework. Derivatives will be used to prune the positive spanning sets considered at the poll step of the algorithm, highlighting the role that ascent directions, that conform to the geometry of the nearby feasible region, can have. Both variants of DMS show to be competitive against a state-of-art derivative-based algorithm. Moreover, for reasonable small budgets of function evaluations, the new variant is not only competitive with the derivative-based solver but also with the original implementation of DMS.
Direct Multisearch (DMS) is a Derivative-free Optimization class of algorithms suited for computing approximations to the complete Pareto front of a given Multiobjective Optimization problem. It has a well-supported convergence analysis and simple implementations present a good numerical performance, both in academic test sets and in real applications. Recently, this numerical performance was improved with the definition of a search step based on the minimization of quadratic polynomial models, corresponding to the algorithm BoostDMS. In this work, we propose and numerically evaluate the perfo…
Direct Multisearch is a well-established class of algorithms, suited for multiobjective derivative-free optimization. In this work, we analyze the worst-case complexity of this class of methods in its most general formulation for unconstrained optimization. Considering nonconvex smooth functions, we show that to drive a given criticality measure below a specific positive threshold, Direct Multisearch takes at most a number of iterations proportional to the square of the inverse of the threshold, raised to the number of components of the objective function. This number is also proportional to t…
The aim of this manuscript is to characterize the continuity properties of the multiobjective steepest descent direction for smooth objective functions. We will show that this direction is Holder continuous with optimal exponent 1/2. In particular, this direction fails to be Lipschitz continuous even for polynomial objectives.
In this paper, we propose a new descent method, termed as multiobjective memory gradient method, for finding Pareto critical points of a multiobjective optimization problem. The main thought in this method is to select a combination of the current descent direction and past multi-step iterative information as a new search direction and to obtain a stepsize by virtue of two types of strategies. It is proved that the developed direction with suitable parameters always satisfies the sufficient descent condition at each iteration. Based on mild assumptions, we obtain the global convergence and the…
In this article we develop a gradient-based algorithm for the solution of multiobjective optimization problems with uncertainties. To this end, an additional condition is derived for the descent direction in order to account for inaccuracies in the gradients and then incorporated in a subdivison algorithm for the computation of global solutions to multiobjective optimization problems. Convergence to a superset of the Pareto set is proved and an upper bound for the maximal distance to the set of substationary points is given. Besides the applicability to problems with uncertainties, the algorit…
The steepest descent method proposed by Fliege et al. motivates the research on descent methods for multiobjective optimization, which has received increasing attention in recent years. However, empirical results show that the Armijo line search often gives a very small stepsize along the steepest direction, which decelerates the convergence seriously. This paper points out that the issue is mainly due to the imbalances among objective functions. To address this issue, we propose a Barzilai-Borwein descent method for multiobjective optimization (BBDMO) that dynamically tunes gradient magnitude…
In this article, we present an efficient descent method for locally Lipschitz continuous multiobjective optimization problems (MOPs). The method is realized by combining a theoretical result regarding the computation of descent directions for nonsmooth MOPs with a practical method to approximate the subdifferentials of the objective functions. We show convergence to points which satisfy a necessary condition for Pareto optimality. Using a set of test problems, we compare our method to the multiobjective proximal bundle method by Mäkelä. The results indicate that our method is competitive while…
We derive efficient algorithms to compute weakly Pareto optimal solutions for smooth, convex and unconstrained multiobjective optimization problems in general Hilbert spaces. To this end, we define a novel inertial gradient-like dynamical system in the multiobjective setting, whose trajectories converge weakly to Pareto optimal solutions. Discretization of this system yields an inertial multiobjective algorithm which generates sequences that converge weakly to Pareto optimal solutions. We employ Nesterov acceleration to define an algorithm with an improved convergence rate compared to the plai…
In this work, we consider multiobjective optimization problems with both bound constraints on the variables and general nonlinear constraints, where objective and constraint function values can only be obtained by querying a black box. Furthermore, we consider the case where a subset of the variables can only take integer values. We propose a new linesearch-based solution method and show that it converges to a set of stationary points for the problem. For what concerns the continuous variables, we employ a strategy for the estimation of the Pareto frontier recently proposed in the literature a…
In a wide range of applications it is desirable to optimally control a dynamical system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a single optimal solution, the set of optimal compromises, the so-called Pareto set, has to be approximated. When the problem under consideration is described by a partial differential equation (PDE), as is the case for fluid flow, the computational cost rapidly increases and makes its direct treatment infeasible. Reduced order modeling is a very popular method to …
This work proposes the integration of two new constraint-handling approaches into the blackbox constrained multiobjective optimization algorithm DMulti-MADS, an extension of the Mesh Adaptive Direct Search (MADS) algorithm for single-objective constrained optimization. The constraints are aggregated into a single constraint violation function which is used either in a two-phase approach, where research of a feasible point is prioritized if not available before improving the current solution set, or in a progressive barrier approach, where any trial point whose constraint violation function val…
The paper concerns multiobjective linear optimization problems in R^n that are parameterized with respect to the right-hand side perturbations of inequality constraints. Our focus is on measuring the variation of the feasible set and the Pareto front mappings around a nominal element while paying attention to some specific directions. This idea is formalized by means of the so-called epigraphical multifunction, which is defined by adding a fixed cone to the images of the original mapping. Through the epigraphical feasible and Pareto front mappings we describe the corresponding vector subdiffer…
We present a flexible trust region descend algorithm for unconstrained and convexly constrained multiobjective optimization problems. It is targeted at heterogeneous and expensive problems, i.e., problems that have at least one objective function that is computationally expensive. The method is derivative-free in the sense that neither need derivative information be available for the expensive objectives nor are gradients approximated using repeated function evaluations as is the case in finite-difference methods. Instead, a multiobjective trust region approach is used that works similarly to …
We present a novel algorithm that allows us to gain detailed insight into the effects of sparsity in linear and nonlinear optimization, which is of great importance in many scientific areas such as image and signal processing, medical imaging, compressed sensing, and machine learning (e.g., for the training of neural networks). Sparsity is an important feature to ensure robustness against noisy data, but also to find models that are interpretable and easy to analyze due to the small number of relevant terms. It is common practice to enforce sparsity by adding the $\ell_1$-norm as a weighted pe…
Multiobjective combinatorial optimization (MOCO) problems can be found in many real-world applications. However, exactly solving these problems would be very challenging, particularly when they are NP-hard. Many handcrafted heuristic methods have been proposed to tackle different MOCO problems over the past decades. In this work, we generalize the idea of neural combinatorial optimization, and develop a learning-based approach to approximate the whole Pareto set for a given MOCO problem without further search procedure. We propose a single preference-conditioned model to directly generate appr…
Benson's outer approximation algorithm and its variants are the most frequently used methods for solving linear multiobjective optimization problems. These algorithms have two intertwined components: one-dimensional linear optimization one one hand, and a combinatorial part closely related to vertex numeration on the other. Their separation provides a deeper insight into Benson's algorithm, and points toward a dual approach. Two skeletal algorithms are defined which focus on the combinatorial part. Using different single-objective optimization problems - called oracle calls - yield different a…
