Scalarization allows to solve a multi-objective optimization problem by solving many single-objective sub-problems, uniquely determined by some parameters. In this work, we propose several adaptive strategies to select such parameters in order to obtain a uniform approximation of the Pareto front. This is done by introducing a heuristic dynamics where the parameters interact through a binary repulsive potential. The approach aims to minimize the associated energy potential which is used to quantify the diversity of the computed solutions. A stochastic component is also added to overcome non-optimal energy configurations. Numerical experiments show the validity of the proposed approach for bi- and tri-objectives problems with different Pareto front geometries.
Selecting the optimal material for a part designed through topology optimization is a complex problem. The shape and properties of the Pareto front plays an important role in this selection. In this paper we show that the compliance-volume fraction Pareto fronts of some topology optimization problems in linear elasticity share some useful properties. These properties provide an interesting point of view on the efficiency of topology optimization compared to other design approaches such as parametric structural optimization. We construct a simple meta-model which requires only one full topology…
Multiobjective Optimization under Uncertainties using Conditional Pareto Fronts
Centrale Lyon, CNRS UMR 5208, Institut Camille Jordan（里昂中央理工学院，法国国家科学研究中心UMR 5208，卡米尔·乔丹研究所）；Mines Saint-Etienne, Univ Clermont Auvergne, INP Clermont Auvergne, CNRS, UMR 6158 LIMOS（圣艾蒂安国立高等矿业学校，克莱蒙奥弗涅大学，克莱蒙奥弗涅国立高等工程师学院，法国国家科学研究中心UMR 6158 LIMOS）；Laboratoire d’Informatique, de Modélisation et d’Optimisation des Systèmes (LIMOS)（系统信息、建模与优化实验室（LIMOS））
Multi-objective integer or mixed-integer programming problems typically have disconnected feasible domains, making the task of constructing an approximation of the Pareto front challenging. The present paper shows that certain algorithms which were originally devised for continuous problems can be successfully adapted to approximate the Pareto front for integer, and mixed-integer, multi-objective optimization problems. Relationships amongst various scalarization techniques are established to motivate the choice of a particular scalarization in these algorithms. The proposed algorithms are test…
In this paper, we consider black-box multiobjective optimization problems in which all objective functions are not given analytically. In multiobjective optimization, it is important to produce a set of uniformly distributed discrete solutions over the Pareto front to build a good approximation. In black-box biobjective optimization, one can evaluate distances between solutions with the ordering property of the Pareto front. These distances allow to be able to evaluate distribution of all solution points, so it is not difficult to maintain uniformity of solutions distribution. However, problem…
Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging task: In the case of three or more objectives the Pareto front is usually difficult to view, if not impossible, and even in the case of just two objectives constructing the whole Pareto front so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or efficient) set has been an active area of research. Although there is a wealth of literature involving finite dimensional…
Dealing with multi-objective problems by using generation methods has some interesting advantages since it provides the decision-maker with the complete information about the set of non-dominated points (Pareto front) and a clear overview of the problem. However, providing many solutions to the decision-maker might also be overwhelming. As an alternative approach, presenting a representative set of solutions of the Pareto front may be advantageous. Choosing such a representative set is by itself also a multi-objective problem that must consider the number of solutions to present, the uniformit…
Hypervolume is widely used in the evolutionary multi-objective optimization (EMO) field to evaluate the quality of a solution set. For a solution set with $μ$ solutions on a Pareto front, a larger hypervolume means a better solution set. Investigating the distribution of the solution set with the largest hypervolume is an important topic in EMO, which is the so-called hypervolume optimal $μ$-distribution. Theoretical results have shown that the $μ$ solutions are uniformly distributed on a linear Pareto front in two dimensions. However, the $μ$ solutions are not always uniformly distributed on …
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Oden Institute for Computational Engineering and Sciences（奥登计算工程与科学研究所）
Classification, recommendation, and ranking problems often involve competing goals with additional constraints (e.g., to satisfy fairness or diversity criteria). Such optimization problems are quite challenging, often involving non-convex functions along with considerations of user preferences in balancing trade-offs. Pareto solutions represent optimal frontiers for jointly optimizing multiple competing objectives. A major obstacle for frequently used linear-scalarization strategies is that the resulting optimization problem might not always converge to a global optimum. Furthermore, such meth…
In this work we are interested in stochastic particle methods for multi-objective optimization. The problem is formulated using parametrized, single-objective sub-problems which are solved simultaneously. To this end a consensus based multi-objective optimization method on the search space combined with an additional heuristic strategy to adapt parameters during the computations is proposed. The adaptive strategy aims to distribute the particles uniformly over the image space by using energy-based measures to quantify the diversity of the system. The resulting metaheuristic algorithm is mathem…
We present Parallel Feasible Pareto Frontier Entropy Search ($\{\text{PF}\}^2$ES) -- a novel information-theoretic acquisition function for multi-objective Bayesian optimization supporting unknown constraints and batch query. Due to the complexity of characterizing the mutual information between candidate evaluations and (feasible) Pareto frontiers, existing approaches must either employ crude approximations that significantly hamper their performance or rely on expensive inference schemes that substantially increase the optimization's computational overhead. By instead using a variational low…
We propose a surrogate-assisted reference vector adaptation (SRVA) method to solve expensive multi- and many-objective optimization problems with various Pareto front shapes. SRVA is coupled with a multi-objective Bayesian optimization (MBO) algorithm using reference vectors for scalarization of objective functions. The Kriging surrogate models for MBO is used to estimate the Pareto front shape and generate adaptive reference vectors uniformly distributed on the estimated Pareto front. We combine SRVA with expected improvement of penalty-based boundary intersection as an infill criterion for M…
In multiobjective optimisation, a set of scalable test problems with a variety of features allow researchers to investigate and evaluate the abilities of different optimisation algorithms, and thus can help them to design and develop more effective and efficient approaches. Existing test problem suites mainly focus on situations where all the objectives are fully conflicting with each other. In such cases, an m-objective optimisation problem has an (m-1)-dimensional Pareto front in the objective space. However, in some optimisation problems, there may be unexpected characteristics among object…
In this paper, to address backhaul capacity bottleneck and concurrently optimize energy consumption and delay, we formulate a novel weighted-sum multi-objective optimization problem where popular content caching placement and integrated access and backhaul (IAB) millimeter (mmWave) bandwidth partitioning are optimized jointly to provide Pareto efficient optimal non-dominating solutions. In such integrated networks analysis of what-if scenarios to understand trade-offs in decision space, without losing sight of optimality, is important. A wide set of numerical investigations reveal that compare…
Kyushu Sangyo University（九州产业大学）；KLab Inc.（KLab公司）；RIKEN（理化学研究所）；Tokyo Institute of Technology（东京工业大学）
A multi-objective optimization problem is $C^r$ weakly simplicial if there exists a $C^r$ surjection from a simplex onto the Pareto set/front such that the image of each subsimplex is the Pareto set/front of a subproblem, where $0\leq r\leq \infty$. This property is helpful to compute a parametric-surface approximation of the entire Pareto set and Pareto front. It is known that all unconstrained strongly convex $C^r$ problems are $C^{r-1}$ weakly simplicial for $1\leq r \leq \infty$. In this paper, we show that all unconstrained strongly convex problems are $C^0$ weakly simplicial. The usefuln…
A Two-Phase Adaptive Balanced Penalty Method for Controllable Pareto Front Learning under Split Feasibility Conditions
Faculty of Applied Mathematics and Informatics, Hanoi University of Science and Technology（应用数学与信息技术学院，河内科学技术大学）；College of Engineering and Computer Science, VinUniversity（工程与计算机科学学院，Vin大学）
We consider the problem of computing the \emph{distance-based representative skyline} in the plane, a problem introduced by Tao, Ding, Lin and Pei [Proc. 25th IEEE International Conference on Data Engineering (ICDE), 2009] and independently considered by Dupin, Nielsen and Talbi [Optimization and Learning - Third International Conference, OLA 2020] in the context of multi-objective optimization. Given a set $P$ of $n$ points in the plane and a parameter $k$, the task is to select $k$ points of the skyline defined by $P$ (also known as Pareto front for $P$) to minimize the maximum distance from…
Optimization of conflicting functions is of paramount importance in decision making, and real world applications frequently involve data that is uncertain or unknown, resulting in multi-objective optimization (MOO) problems of stochastic type. We study the stochastic multi-gradient (SMG) method, seen as an extension of the classical stochastic gradient method for single-objective optimization. At each iteration of the SMG method, a stochastic multi-gradient direction is calculated by solving a quadratic subproblem, and it is shown that this direction is biased even when all individual gradient…
Lehigh University（理海大学）；The Ohio State University（俄亥俄州立大学）；Drake University（德雷克大学）；Cornell University（康奈尔大学）
Multiple-objective optimization (MOO) aims to simultaneously optimize multiple conflicting objectives and has found important applications in machine learning, such as minimizing classification loss and discrepancy in treating different populations for fairness. At optimality, further optimizing one objective will necessarily harm at least another objective, and decision-makers need to comprehensively explore multiple optima (called Pareto front) to pinpoint one final solution. We address the efficiency of finding the Pareto front. First, finding the front from scratch using stochastic multi-g…
Pareto optimization of resonances and minimum-time control
本文的目的是减少一个频谱最优化问题，这涉及到最小化的衰减率$|\mathrm{Im} \, k |$的共振$k$，收集的最优控制问题的黎曼球$\widehat{\mathbb{C}}$。这种减少使我们能够应用极值合成的方法，分层光学腔的结构优化。我们从一个对偶问题的谐振器长度的最小化，并给出了几个这个问题，涉及帕累托优化的模量$|k|$的谐振，最小时间控制问题的$\widehat{\mathbb{C}}$，和相关的哈密尔顿-雅可比-贝尔曼方程。各种类型的可控性研究与优化器的存在和最小衰减和最小模的帕累托最优边界之间的关系。我们给出明确的例子，最佳共振和定性描述他们附近的帕累托边界的属性。一个特殊的表示的bang-bang控制轨迹相结合的极值分析，以获得各种界限的最佳层的宽度。提出了一种基于最小时间控制的最优对称谐振器的计算方法，并以高精度计算了几个Pareto最优前沿和高Q值谐振器。
