A derivative-free approach to mixed integer constrained multiobjective nonsmooth black-box optimization

The context of this research is multiobjective optimization where conflicting objectives are present. In this work, these objectives are only available as the outputs of a blackbox for which no derivative information is available. This work proposes a new extension of the mesh adaptive direct search (MADS) algorithm to multiobjective derivative-free optimization with bound constraints. This method does not aggregate objectives and keeps a list of non dominated points which converges to a (local) Pareto set as long as the algorithm unfolds. As in the single-objective optimization MADS algorithm, this method is built around a search step and a poll step. Under classical direct search assumptions, it is proved that the so-called DMulti-MADS algorithm generates multiple subsequences of iterates which converge to a set of local Pareto stationary points. Finally, computational experiments suggest that this approach is competitive compared to the state-of-the-art algorithms for multiobjective blackbox optimization.

In this paper, we investigate a robust nonsmooth multiobjective optimization problem related to a multiobjective optimization with data uncertainty. We firstly introduce two kinds of generalized convex functions, which are not necessary to be convex. Robust necessary optimality conditions for weakly robust efficient solutions and properly robust efficient solutions of the problem are established by a generalized alternative theorem and the robust constraint qualification. Further, robust sufficient optimality conditions for weakly robust efficient solutions and properly robust efficient solutions of the problem are also derived. The Mond–Weir-type dual problem and Wolfe-type dual problem are formulated. Finally, we obtain the weak, strong and converse robust duality results between the primal one and its dual problems under the generalized convexity assumptions.

A proximal point method for nonsmooth multiobjective optimization in the Riemannian context is proposed, and an optimality condition for multiobjective problems is introduced. This allowed replacing the classic approach, via “scalarization,” by a purely vectorial and considering the method without any assumption of convexity over the constraint sets that determine the vectorial improvement steps. The main convergence result ensures that each cluster point (if any) of any sequence generated by the method is a Pareto critical point. Moreover, when the problem is convex on a Hadamard manifold, full convergence of the method for a weak Pareto optimal is obtained.

Multi-objective operation management of a multi-carrier energy system

A generalization of the cubic algorithm is presented for global optimization of nonconvex nonsmooth multiobjective optimization programs $\min f_{s}(x), s=1,\dots,k,$ with box constraints $x\in X=[a_{1},b_{1}]\times \dots\times\lbrack a_{n},b_{n}]$. &nbsp &nbsp This monotonic set contraction algorithm converges onto the entire exact Pareto set, if nonempty, and yields its approximation with given precision in a finite number of iterations. Simultaneously, approximations for the ideal point and for the function values over Pareto set are obtained. The method is implemented by Maple code, and this code does not create ill-conditioned situations. &nbsp &nbsp Results of numerical experiments are presented, with graphs, to illustrate the use of the code, and the solution set can be visualized in projections on coordinate planes. The code is ready for engineering and economic applications.

We study black-box vector optimization with Gaussian process bandits, where there is an incomplete order relation on objective vectors described by a polyhedral convex cone. Existing black-box vector optimization approaches either suffer from high sample complexity or lack theoretical guarantees. We propose Vector Optimization with Gaussian Process (VOGP), an adaptive elimination algorithm that identifies Pareto optimal solutions sample efficiently by exploiting the smoothness of the objective function. We establish theoretical guarantees, deriving information gain-based and kernel-specific sample complexity bounds. Finally, we conduct a thorough empirical evaluation of VOGP and compare it with the state-of-the-art multi-objective and vector optimization algorithms on several real-world and synthetic datasets, emphasizing VOGP’s efficiency ( e.g. , $$\sim 18\times$$ lower sample complexity on average). We also provide heuristic adaptations of VOGP for cases where the design space is continuous and where the Gaussian process model lacks access to the true kernel hyperparameters. This work opens a new frontier in sample-efficient multi-objective black-box optimization by incorporating preference structures while maintaining theoretical guarantees and practical efficiency.

Many evolutionary algorithms are designed to solve black-box multi-objective optimization problems MOPs using stochastic operators, where neither the form nor the gradient information of the problem is accessible. In some real-world applications, e.g. surrogate-based global optimization, the gradient of the objective function is accessible. In this case, it is straightforward to use a gradient-based multi-objective optimization algorithm to achieve fast convergence speed and the stability of the solution. In a relatively recent approach, the hypervolume indicator gradient in the decision space is derived, which paves the way for the method for maximizing the hypervolume indicator of a fixed size population. In this paper, several mechanisms which originated in the field of evolutionary computation are proposed to make this gradient ascent method applicable. Specifically, the well-known non-dominated sorting is used to help steering the dominated points. The principle of the so-called cumulative step-size control that is originally proposed for evolution strategies is adapted to control the step-size dynamically. The resulting algorithm is called Hypervolume Indicator Gradient Ascent Multi-objective Optimization HIGA-MO. The proposed algorithm is tested on ZDT problems and its performance is compared to other methods of moving the dominated points as well as to some evolutionary multi-objective optimization algorithms that are commonly used.

A mesh adaptive direct search algorithm for multiobjective optimization

Groundwater overpumping could cause serious land subsidence. Although groundwater management models have been widely applied to obtain optimal pumping strategies for land subsidence control, most of them have not explicitly incorporated the land subsidence into model’s constraints. This study presented a groundwater management model explicitly considering land subsidence. To quantify the relation between land subsidence and drawdown, the one-dimensional consolidation equation was adopted which simultaneously accounts for the elastic and inelastic compaction due to pumping. Based on the response matrix technique along with one-dimensional consolidation equation, a groundwater quantity management model was developed which enables the determination of maximum total pumpage subject that the land subsidence do not exceed the allowable value. The developed management model was non-smooth optimization problem. To improve the solution efficiency, the non-smooth optimization was transferred into mixed integer linear programming (MILP) by introducing the binary variables. A hypothetical example was utilized to demonstrate the applicability of developed model. The results indicated that the land subsidence should be explicitly incorporated into model’s constraints; otherwise the optimal total pumpage might be overestimated. Moreover, the simultaneous consideration of elastic and inelastic compaction in groundwater management was important, especially when the difference between initial head and preconsolidation head was significant.

Multi Objective Optimization Using Life Cycle Environmental Impact and Cost in the Operation of Utility Plants

Methods for multi-objective investment and operating optimization of complex energy systems

Human-robot collaborative disassembly lines are widely used by remanufacturing companies to disassemble end-of-life (EOL) products. When disassembling large-sized EOL products, each workstation on a disassembly line is generally divided into multiple operating positions, so that different operators can disassemble the same product at their respective positions at the same time, thereby greatly improving efficiency. This paper focuses on a position-constrained human-robot collaborative disassembly planning (PC-HRCDP) problem for the above-mentioned lines, including three subproblems of disassembly sequence planning, disassembly line balancing and robot path planning. A multi-objective mixed integer programming model for PC-HRCDP is developed to solve small-scale instances. Furthermore, a multi-objective multi-fidelity optimisation (MO-MFO) algorithm is proposed to solve large-scale instances. Comprehensive experiments are conducted based on 10 problem instances generated in this study. Experimental results show that the proposed MO-MFO is better than a high-fidelity optimisation algorithm in terms of running time. In addition, benefiting from the strategy of MO-MFO to allocate the limited high-fidelity computational budget to solutions in the two stages of multi-objective optimisation and optimal sampling, MO-MFO is significantly better than the existing representative multi-fidelity optimisation algorithms in terms of the hyper-volume and the inverted generational distance.

In this paper second-order necessary optimality conditions for nonsmooth vector optimization problems are given by smooth approximations. We extend to the vector case the approach introduced by Ermoliev, Norkin and Wets to define generalized derivatives for discontinuous functions as limit of the classical derivatives of regular functions.KeywordsVector optimizationOptimality conditionsMollifiersTaylor’s Formula

Using the idea of upper convexificators, we propose constraint qualifications and study existence and boundedness of the Kuhn-Tucker multipliers for a nonsmooth multiobjective optimization problem with inequality constraints and an arbitrary set constraint. We show that, at locally weak efficient solutions where the objective and constraint functions are locally Lipschitz, the constraint qualifications are necessary and sufficient conditions for the Kuhn-Tucker multiplier sets to be nonempty and bounded under certain semiregularity assumptions on the upper convexificators of the functions.

Saddle-point criteria and Wolfe-type duality for LU-$$(\Phi ,\rho )$$-approximated nonsmooth multiobjective optimization problems with applications in energy management systems