Complexity of Geometric Programming in the Turing Model and Application to Nonnegative Tensors

Kernekoncepter

Efficient computation of minimizers for geometric programming problems.

Resumé

The content discusses the complexity of geometric programming in the Turing model and its application to nonnegative tensors. It covers coerciveness conditions, minimal points, iteration complexity bounds, and poly-time computation methods. The results provide insights into approximating spectral radii and maximum values efficiently.

Introduction:

Multidimensional arrays with d ⩾ 3 indices are prevalent in various fields.

Notation and preliminary results:

Cone of homogeneous polynomials with nonnegative coefficients.
Homogeneous polynomial maps.
Minimization of max of log-Laplace transforms.

Coerciveness condition:

Conditions for compact sublevel sets.

Minimal points:

Hessian interpretation and properties of minimizers.

A bound on Kmin(f) for coercive f:

Lower estimate of ν(A) based on set properties.

Poly-time computation of fmin:

Theorem on efficient computation with given precision ε.

Statistik

Under a coercive assumption, an ε-minimizer can be computed in poly-time.
The spectral radius can be approximated within ε error in poly-time.

Citater

Vigtigste indsigter udtrukket fra

Complexity of Geometric programming in the Turing model and application to nonnegative tensors

by Shmu... kl. arxiv.org 03-20-2024

https://arxiv.org/pdf/2301.10637.pdf

Dybere Forespørgsler

How does the ellipsoid method contribute to finding minimizers efficiently

The ellipsoid method plays a crucial role in efficiently finding minimizers by iteratively refining the search space to converge towards the optimal solution. It involves enclosing the feasible region within an ellipsoidal shape and updating this shape based on feedback from separation oracles until it tightly encapsulates the minimum point. By utilizing weak separation oracles, which provide information about how close a given point is to the feasible region, the ellipsoid method can effectively navigate towards the optimum while minimizing computational overhead.

What implications do these results have for computational complexity theory

These results have significant implications for computational complexity theory as they demonstrate polynomial-time computability of minimization problems that fall under geometric programming. The ability to approximate solutions within a specified error margin in poly-time showcases advancements in solving complex optimization tasks efficiently. Moreover, by establishing bit-size estimates and applying methods like the ellipsoid algorithm, these findings contribute to understanding and addressing challenges related to NP-hard problems in various domains.

How can these findings be applied to other optimization problems beyond geometric programming

The findings presented regarding geometric programming and efficient minimization techniques can be extended to other optimization problems beyond their current scope. The methodologies developed, such as coerciveness conditions, bit-size estimates, and poly-time approximation algorithms, are fundamental principles that can be adapted and applied across different optimization scenarios. For instance, these approaches could be utilized in convex optimization problems, machine learning algorithms requiring efficient convergence strategies, or even resource allocation models where quick decision-making processes are essential for optimal outcomes. By leveraging similar techniques and frameworks derived from geometric programming complexities, researchers can enhance problem-solving capabilities across diverse fields requiring optimization solutions.