Several test function suites are being used for numerical benchmarking of multiobjective optimization algorithms. While they have some desirable properties, such as well-understood Pareto sets and Pareto fronts of various shapes, most of the currently used functions possess characteristics that are arguably underrepresented in real-world problems such as separability, optima located exactly at the boundary constraints, and the existence of variables that solely control the distance between a solution and the Pareto front. Via the alternative construction of combining existing single-objective problems from the literature, we describe the bbob-biobj test suite with 55 bi-objective functions in continuous domain, and its extended version with 92 bi-objective functions ( bbob-biobj-ext ). Both test suites have been implemented in the COCO platform for black-box optimization benchmarking and various visualizations of the test functions are shown to reveal their properties. Besides providing details on the construction of these problems and presenting their (known) properties, this article also aims at giving the rationale behind our approach in terms of groups of functions with similar properties, objective space normalization, and problem instances. The latter allows us to easily compare the performance of deterministic and stochastic solvers, which is an often overlooked issue in benchmarking.

This paper analyzes a -Evolution Strategy, a randomized comparison-based adaptive search algorithm optimizing a linear function with a linear constraint. The algorithm uses resampling to handle the constraint. Two cases are investigated: first, the case where the step-size is constant, and second, the case where the step-size is adapted using cumulative step-size adaptation. We exhibit for each case a Markov chain describing the behavior of the algorithm. Stability of the chain implies, by applying a law of large numbers, either convergence or divergence of the algorithm. Divergence is the desired behavior. In the constant step-size case, we show stability of the Markov chain and prove the divergence of the algorithm. In the cumulative step-size adaptation case, we prove stability of the Markov chain in the simplified case where the cumulation parameter equals 1, and discuss steps to obtain similar results for the full (default) algorithm where the cumulation parameter is smaller than 1. The stability of the Markov chain allows us to deduce geometric divergence or convergence, depending on the dimension, constraint angle, population size, and damping parameter, at a rate that we estimate. Our results complement previous studies where stability was assumed.