Leximin is a common approach to multi-objective optimization, frequently employed in fair division applications. In leximin optimization, one first aims to maximize the smallest objective value; subject to this, one maximizes the second-smallest objective; and so on. Often, even the single-objective problem of maximizing the smallest value cannot be solved accurately. What can we hope to accomplish for leximin optimization in this situation? Recently, Henzinger et al. (2022) defined a notion of \emph{approximate} leximin optimality. Their definition, however, considers only an additive approximation. In this work, we first define the notion of approximate leximin optimality, allowing both multiplicative and additive errors. We then show how to compute, in polynomial time, such an approximate leximin solution, using an oracle that finds an approximation to a single-objective problem. The approximation factors of the algorithms are closely related: an $(\alpha,\epsilon)$-approximation for the single-objective problem (where $\alpha \in (0,1]$ and $\epsilon \geq 0$ are the multiplicative and additive factors respectively) translates into an $\left(\frac{\alpha^2}{1-\alpha + \alpha^2}, \frac{\epsilon}{1-\alpha +\alpha^2}\right)$-approximation for the multi-objective leximin problem, regardless of the number of objectives.

As "a new frontier in evolutionary computation research", evolutionary transfer optimization(ETO) will overcome the traditional paradigm of zero reuse of related experience and knowledge from solved past problems in researches of evolutionary computation. In scheduling applications via ETO, a quite appealing and highly competitive framework "meeting" between them could be formed for both intelligent scheduling and green scheduling, especially for international pledge of "carbon neutrality" from China. To the best of our knowledge, our paper on scheduling here, serves as the 1st work of a class of ETO frameworks when multiobjective optimization problem "meets" single-objective optimization problems in discrete case (not multitasking optimization). More specifically, key knowledge conveyed for industrial applications, like positional building blocks with genetic algorithm based settings, could be used via the new core transfer mechanism and learning techniques for permutation flow shop scheduling problem(PFSP). Extensive studies on well-studied benchmarks validate firm effectiveness and great universality of our proposed ETO-PFSP framework empirically. Our investigations (1) enrich the ETO frameworks, (2) contribute to the classical and fundamental theory of building block for genetic algorithms and memetic algorithms, and (3) head towards the paradigm shift of evolutionary scheduling via learning by proposal and practice of paradigm of "knowledge and building-block based scheduling" (KAB2S) for "industrial intelligence" in China.

Evolutionary transfer optimization(ETO) serves as "a new frontier in evolutionary computation research", which will avoid zero reuse of experience and knowledge from solved problems in traditional evolutionary computation. In scheduling applications via ETO, a highly competitive "meeting" framework between them could be constituted towards both intelligent scheduling and green scheduling, especially for carbon neutrality within the context of China. To the best of our knowledge, our study on scheduling here, is the 1st work of ETO for complex optimization when multiobjective problem "meets" single-objective problems in combinatorial case (not multitasking optimization). More specifically, key knowledge like positional building blocks clustered, could be learned and transferred for permutation flow shop scheduling problem (PFSP). Empirical studies on well-studied benchmarks validate relatively firm effectiveness and great potential of our proposed ETO-PFSP framework.

Whether it is in the field of production, logistics, in medicine or biology; everywhere the global optimal solution or the set of global optimal solutions is sought. However, most real-world problems are of nonlinear nature and naturally multimodal which poses severe problems to global optimization. Multimodality, the existence of multiple (local) optima, is regarded as one of the biggest challenges for continuous single-objective problems [23]. A lot of algorithms get stuck searching for the global optimum or are requiring many function evaluations to escape local optima. One of the most popular strategies for dealing with multimodal problems are population-based methods like evolutionary algorithms due to their global search abilities [2]. In this paper we will examine another approach of coping with local traps, namely multiobjectivization. By transforming a single-objective into a multi-objective problem, we aim at exploiting the properties of multi-objective landscapes. So far, the characteristics of single-objective optimization problems have often been directly transferred to the multiobjective domain.

Before moving on, let's take some time to have a closer look at a single-objective problem. This will give us some perspective. In single-objective problems, the objective is to find a single solution which represents the global optimum in the entire search space. Determining which solutions outperforms others is a simple task when only considering a single-objective, because the best solution is simply the one with the highest (for maximisation problems) or lowest (for minimisation problems) objective value. Let's take the Sphere function as an example.

Multi-objective problems with correlated objectives are a class of problems that deserve specific attention. In contrast to typical multi-objective problems, they do not require the identification of trade-offs between the objectives, as (near-) optimal solutions for any objective are (near-) optimal for every objective. Intelligently combining the feedback from these objectives, instead of only looking at a single one, can improve optimization. This class of problems is very relevant in reinforcement learning, as any single-objective reinforcement learning problem can be framed as such a multi-objective problem using multiple reward shaping functions. After discussing this problem class, we propose a solution technique for such reinforcement learning problems, called adaptive objective selection. This technique makes a temporal difference learner estimate the Q-function for each objective in parallel, and introduces a way of measuring confidence in these estimates. This confidence metric is then used to choose which objective's estimates to use for action selection. We show significant improvements in performance over other plausible techniques on two problem domains. Finally, we provide an intuitive analysis of the technique's decisions, yielding insights into the nature of the problems being solved.