Tropical algebra for noise removal and optimal control

Abstract

Algorithms for noise removal are either complex or ineffective, and the optimal control with inequality constrains makes the algorithm even more complex. Now the condition is changed completely, and the tropical algebra is an extremely simple tool for this purpose. The tropical algebra-based filter has obvious advantages over traditional ones, and an inequality constrain can be converted to a tropical polynomial, making the tropical algebra much attractive in engineering applications. In this paper, idempotents on multiplicative semigroups of tropical matrices are studied. First, the concepts of tropical algebra and the semigroup of tropical matrices under multiplication are introduced. Second, the structure of idempotents on the semigroup of [Formula: see text] tropical matrices under multiplication is given. Finally, as examples, the tropical addition is used as a filter for noise removal, and a simple optimization is given where the constraint is given by the tropical algebra. The paper has opened the path for a new way to noise removal and optimization.

Tropical algebra with high-order matrix for multiple-noise removal

“…He proved that the upper triangular tropical matrix semigroup UT n ðTÞ generates a different semigroup variety for each dimension n. In 2022, Kambites 35 studied the free objects in the variety of semigroups and variety of monoids generated by the monoid of all n × n upper triangular matrices over a commutative semiring. In 2022, Gong et al 18 studied idempotents on multiplicative semigroups of 3 × 3 tropical matrices, and the structure of idempotents on this kind of semigroup is given. We call a is an idempotent of semigroup S when a 2 ¼ a.…”

“…In 2022, Kambites 35 studied the free objects in the variety of semigroups and variety of monoids generated by the monoid of all n×n upper triangular matrices over a commutative semiring. In 2022, Gong et al 18 studied idempotents on multiplicative semigroups of 3×3 tropical matrices, and the structure of idempotents on this kind of semigroup is given.…”

“…17 A typical denoising problem is the combination of tropical algebra with denoising and optimal control. In 2022, Gong et al 18 introduced tropical algebra into noise removal, used tropical addition as filter denoising, transformed inequality constraints in traditional algorithms into tropical polynomials, and presented a simple optimization method with constraints given by tropical algebra. Gong pointed out that in the sense of tropical algebra, the traditional optimization problem can be solved by tropical polynomials.…”

A Family of Multi-Step Subgradient Minimization Methods

“…For a more detailed review of conjugate gradient methods, see [40,53]. It seems relevant to create multi-step universal methods for solving non-smooth problems that are applicable in terms of computer memory resources for solving highdimensional minimization problems [54][55][56][57]. In this work, we propose a family of multistep RSMs for solving large-scale problems.…”