## 1 Introduction
As far as other optimization methods are concerned, [PERSON] and [PERSON] (2023) [13], enhanced the BESO (Bi-directional Evolutionary Structural Optimization) method by incorporating buckling constraints, effectively addressing pseudo buckling modes in structural topology optimization; [PERSON] (2020) developed an element-based topology optimization algorithm for a smooth design [14] that was further investigated in [PERSON] and [PERSON] (2022), [15], that introduced a three-field FPTO (Floating Projection Topology Optimization) method employing linear material interpolation, facilitating the generation of clear 0/1 designs without the need for penalization schemes. [PERSON] et al. (2020) [16], developed the SEMDOT (Smooth-Edged Material Distribution for Optimizing Topology) algorithm, which effectively produces optimized topologies with smooth and clear boundaries, outperforming traditional methods in certain applications. Second, controlling the interface between material phases--whether through filtering, projection, or constraint aggregation--proves essential in achieving physically meaningful designs. Third, scalability and code accessibility, as emphasized by [PERSON] et al. [6] and [PERSON] [17], are key to enabling broader adoption of MMTO by researchers and practitioners. Despite these advances, MMTO remains a vibrant and rapidly evolving field. Open challenges include the integration of nonlinear material behaviors, uncertainty quantification, multi-physics coupling, and automated manufacturability constraints. Moreover, emerging fabrication technologies continue to push the boundaries of what is realizable, demanding increasingly sophisticated optimization formulations. Given this background, the present paper proposes an innovative formulation for the topology optimization of Multi-Input (multi-loaded) Multi-Output (multi-objective) structures in a multi-material setting. While the framework naturally includes classical compliance optimization as a special case, it is sufficiently general to accommodate alternative compliance definitions and other objective functions tailored to specific applications. The central idea behind the proposed approach is the explicit formulation of the input-output transfer matrix, denoted as \(\mathbf{G}\), which maps applied loads to the structural response. In this framework, the structural response vector--often referred to as the output vector--can be any linear combination of the state vector components. Notably, structural compliance, which is the primary objective in most topology optimization studies, can inherently be expressed as a linear combination of displacements at the locations where loads are applied. The applied loads themselves serve as coefficients of this combination, as demonstrated later in this paper. Consequently, the optimization problem is formulated to minimize a suitable measure of the \"amplitude\" of the transfer matrix \(\mathbf{G}\), which, from an engineering perspective, corresponds to reducing the impact of applied loads on the structure. A key component of this methodology is the selection of an appropriate metric to quantify the amplitude of the transfer matrix. The Singular Value Decomposition (SVD) [18; 19] emerges as a natural tool for this purpose. Not only does SVD facilitate the computation of matrix norms that serve as amplitude measures, but it also provides valuable insights into system behavior. Furthermore, since these norms are uniquely defined in terms of the singular values of \(\mathbf{G}\), the computation of sensitivities is significantly simplified--reducing the problem to evaluating the gradients of singular values with respect to design parameters. The formulation presented in this paper extends to MMTO the one presented in [20] that is restricted to standard topology optimization in a SIMP framework. As for the 3D idealization, the formulation introduced in [21] shall be extended to the multimaterial case in a forthcoming contribution. The remainder of this paper is structured as follows.