We focus on kernel methods for set-valued inputs and their application to Bayesian set optimization, notably combinatorial optimization. We investigate two classes of set kernels that both rely on Reproducing Kernel Hilbert Space embeddings, namely the "Double Sum" (DS) kernels recently considered in Bayesian set optimization, and a class introduced here called "Deep Embedding" (DE) kernels that essentially consists in applying a radial kernel on Hilbert space on top of the canonical distance induced by another kernel such as a DS kernel. We establish in particular that while DS kernels typically suffer from a lack of strict positive definiteness, vast subclasses of DE kernels built upon DS kernels do possess this property, enabling in turn combinatorial optimization without requiring to introduce a jitter parameter. Proofs of theoretical results about considered kernels are complemented by a few practicalities regarding hyperparameter fitting. We furthermore demonstrate the applicability of our approach in prediction and optimization tasks, relying both on toy examples and on two test cases from mechanical engineering and hydrogeology, respectively. Experimental results highlight the applicability and compared merits of the considered approaches while opening new perspectives in prediction and sequential design with set inputs.

This paper presents a parallel surrogate-based global optimization method for computationally expensive objective functions that is more effective for larger numbers of processors. To reach this goal, we integrated concepts from multi-objective optimization and tabu search into, single objective, surrogate optimization. Our proposed derivative-free algorithm, called SOP, uses non-dominated sorting of points for which the expensive function has been previously evaluated. The two objectives are the expensive function value of the point and the minimum distance of the point to previously evaluated points. Based on the results of non-dominated sorting, P points from the sorted fronts are selected as centers from which many candidate points are generated by random perturbations. Based on surrogate approximation, the best candidate point is subsequently selected for expensive evaluation for each of the P centers, with simultaneous computation on P processors. Centers that previously did not generate good solutions are tabu with a given tenure. We show almost sure convergence of this algorithm under some conditions. The performance of SOP is compared with two RBF based methods. The test results show that SOP is an efficient method that can reduce time required to find a good near optimal solution. In a number of cases the efficiency of SOP is so good that SOP with 8 processors found an accurate answer in less wall-clock time than the other algorithms did with 32 processors.

SO-MODS is a new algorithm that combines surrogate global optimization methods with local search. SO-MODS is an extension of prior algorithms that sought to find near optimal solutions for computationally very expensive functions for which the number of allowable evaluations is strictly limited. The global search method in SO-MODS perturbs the best point found so far in order to find a new sample point. The number of decision variables being perturbed is dynamically adjusted in each iteration in order to be more effective for higher dimensional problems. The procedure for dynamically changing the dimensions perturbed is drawn from earlier work on the DYCORS algorithm. We use a cubic radial basis function as surrogate model and investigate two approaches to improve the solution accuracy. The numerical results show that SO-MODS is able to reduce the objective function value dramatically with just a few hundred evaluations even for 30-dimensional problems. The local search is then able to reduce the objective function value further.