[Paper Review] Idempotent interval analysis and optimization problems

Grigori Litvinov, Andreĭ Sobolevskiĭ|ArXiv.org|Jan 10, 2001

Numerical Methods and Algorithms17 references10 citations

TL;DR

This paper introduces idempotent interval analysis, a novel framework for solving optimization problems with interval uncertainties by leveraging the algebraic structure of idempotent semirings—particularly the max-plus algebra. It demonstrates that interval systems in this setting can be solved in polynomial time, unlike traditional interval linear algebra, which is NP-hard, and applies the method to the discrete stationary Bellman equation with exact interval solutions.

ABSTRACT

Many problems in optimization theory are strongly nonlinear in the traditional sense but possess a hidden linear structure over suitable idempotent semirings. After an overview of `Idempotent Mathematics' with an emphasis on matrix theory, interval analysis over idempotent semirings is developed. The theory is applied to construction of exact interval solutions to the interval discrete stationary Bellman equation. Solution of an interval system is typically NP-hard in the traditional interval linear algebra; in the idempotent case it is polynomial. A generalization to the case of positive semirings is outlined.

Motivation & Objective

• To develop a theoretical framework for interval analysis over idempotent semirings, extending classical interval arithmetic to non-standard algebraic structures.
• To address the computational intractability of traditional interval linear systems (typically NP-hard) by exploiting hidden linearity in idempotent semirings.
• To provide exact interval solutions to the discrete stationary Bellman equation, a central problem in discrete optimization with uncertain inputs.
• To generalize the results to positive semirings, broadening applicability beyond idempotent structures.
• To establish that key operations—such as matrix multiplication and solution of linear systems—remain associative and distributive in the idempotent setting, unlike in classical interval arithmetic.

Proposed method

• Formalizing interval arithmetic over idempotent semirings, particularly the max-plus algebra (R_max), where addition is max and multiplication is standard addition.
• Defining weak and strong interval extensions of positive semirings, and introducing interval matrices with component-wise ordering.
• Establishing that matrix operations in the idempotent interval setting preserve associativity and distributivity, unlike classical interval arithmetic.
• Applying the idempotent superposition principle to reformulate nonlinear optimization problems as linear systems over idempotent semirings.
• Using the closure operation * in positive semirings to define the algebraic solution X = A*X ⊕ B, and proving that the interval matrix A*B contains all possible solutions.
• Proving that the interval solution A*B is the minimal interval containing all solutions, using monotonicity and positivity of the semiring.

Experimental results

Research questions

• RQ1 Can interval analysis over idempotent semirings provide a computationally tractable alternative to classical interval linear algebra for solving optimization problems?
• RQ2 To what extent does the idempotent structure preserve desirable algebraic properties (e.g., associativity, distributivity) that are lost in classical interval arithmetic?
• RQ3 Can the interval discrete Bellman equation be solved exactly using this framework, and what is the computational complexity of such a solution?
• RQ4 How does the solution set of an interval linear system in the idempotent setting compare to that in the classical setting in terms of complexity and structure?
• RQ5 Can the results be generalized beyond idempotent semirings to broader classes of positive semirings, such as nonnegative reals with standard operations?

Key findings

• Interval systems in the idempotent setting can be solved in polynomial time, in contrast to the NP-hard complexity of classical interval linear systems.
• The interval matrix solution A*B, where A and B are interval matrices and * denotes the closure operation, contains all possible solutions to the equation X = AX ⊕ B.
• The interval solution A*B is the minimal such interval containing all solutions, making it an exact and optimal enclosure.
• The framework preserves associativity and distributivity of matrix operations, which are lost in classical interval arithmetic, due to monotonicity and positivity in the semiring.
• The solution to the interval discrete stationary Bellman equation is exact and computable in polynomial time using this idempotent interval analysis.
• The results generalize to positive semirings, including the nonnegative reals, with analogous solution properties and minimal interval enclosures.