Game formulations for structural optimization under uncertainty

Abstract

Summary We consider structural optimization (SO) under uncertainty formulated as a mathematical game between two players –– a “designer” and “nature”. The first player wants to design a structure that performs optimally, whereas the second player tries to find the worst possible conditions to impose on the structure. Several solution concepts exist for such games, including Stackelberg and Nash equilibria and Pareto optima. Pareto optimality is shown not to be a useful solution concept. Stackelberg and Nash games are, however, both of potential interest, but these concepts are hardly ever discussed in the literature on SO under uncertainty. Based on concrete examples of topology optimization of trusses and finite element‐discretized continua under worst‐case load uncertainty, we therefore analyze and compare the two solution concepts. In all examples, Stackelberg equilibria exist and can be found numerically, but for some cases we demonstrate nonexistence of Nash equilibria. This motivates a view of the Stackelberg solution concept as the correct one. However, we also demonstrate that existing Nash equilibria can be found using a simple so‐called decomposition algorithm, which could be of interest for other instances of SO under uncertainty, where it is difficult to find a numerically efficient Stackelberg formulation.

Topology optimization with local stress constraints and continuously varying load direction and magnitude: towards practical applications

“…Holmberg et al [27] proposed a game theory approach to solve the worst-case-oriented stress constraints problems considering variation in the load direction; however, because the problems are non-convex, the existence of Nash equilibrium to the game problem proposed by Holmberg et al [27] is not guaranteed, and they rely on empirical observation of the numerical results. In the subsequent work, Thore et al [28] modelled the problem as a Stackelberg game, instead of a zero-sum game, and showed that, with this interpretation, it is possible to guarantee the existence of a solution to the problem. However, the numerical methodology to solve the Stackelberg game problem can be computationally inefficient.…”

“…In the subsequent work, Thore et al. [28] modelled the problem as a Stackelberg game, instead of a zero-sum game, and showed that, with this interpretation, it is possible to guarantee the existence of a solution to the problem. However, the numerical methodology to solve the Stackelberg game problem can be computationally inefficient.…”

A mathematical game for topology optimization of cooling systems

“…Our design problem is a mathematical game [5, 20, 57] between two players, each seeking to minimize an objective directly associated with either flow or temperature by controlling the design.…”

“…Our design problem is a mathematical game [5,20,57] between two players, each seeking to minimize an objective directly associated with either flow or temperature by controlling the design. The objective of the first player is to minimize the fluid compliance, which since we have no nonzero prescribed velocities amounts to minimizing…”

Análise De Viabilidade Técnica De Utilização Da Fibra De Bananeira Com Resina Sintética Em Compósitos

“…Bucher (2018) discussed strategies to have a suitable response surface model, evaluating its quality prediction, and discerning about important and unimportant variables. Thore et al (2019) proposed a topology structural optimization with uncertainty consideration, comparing three solution concepts: Pareto optima, Stackelberg equilibria, and Nash equilibria. Zhao et al (2020) introduced a novel approach based on the graph theory and the set theory aiming at controlling the number and size of interior holes of the structures.…”