12,126 research outputs found
Smaller SDP for SOS Decomposition
A popular numerical method to compute SOS (sum of squares of polynomials)
decompositions for polynomials is to transform the problem into semi-definite
programming (SDP) problems and then solve them by SDP solvers. In this paper,
we focus on reducing the sizes of inputs to SDP solvers to improve the
efficiency and reliability of those SDP based methods. Two types of
polynomials, convex cover polynomials and split polynomials, are defined. A
convex cover polynomial or a split polynomial can be decomposed into several
smaller sub-polynomials such that the original polynomial is SOS if and only if
the sub-polynomials are all SOS. Thus the original SOS problem can be
decomposed equivalently into smaller sub-problems. It is proved that convex
cover polynomials are split polynomials and it is quite possible that sparse
polynomials with many variables are split polynomials, which can be efficiently
detected in practice. Some necessary conditions for polynomials to be SOS are
also given, which can help refute quickly those polynomials which have no SOS
representations so that SDP solvers are not called in this case. All the new
results lead to a new SDP based method to compute SOS decompositions, which
improves this kind of methods by passing smaller inputs to SDP solvers in some
cases. Experiments show that the number of monomials obtained by our program is
often smaller than that by other SDP based software, especially for polynomials
with many variables and high degrees. Numerical results on various tests are
reported to show the performance of our program.Comment: 18 page
Constructing Fewer Open Cells by GCD Computation in CAD Projection
A new projection operator based on cylindrical algebraic decomposition (CAD)
is proposed. The new operator computes the intersection of projection factor
sets produced by different CAD projection orders. In other words, it computes
the gcd of projection polynomials in the same variables produced by different
CAD projection orders. We prove that the new operator still guarantees
obtaining at least one sample point from every connected component of the
highest dimension, and therefore, can be used for testing semi-definiteness of
polynomials. Although the complexity of the new method is still doubly
exponential, in many cases, the new operator does produce smaller projection
factor sets and fewer open cells. Some examples of testing semi-definiteness of
polynomials, which are difficult to be solved by existing tools, have been
worked out efficiently by our program based on the new method.Comment: Accepted by ISSAC 2014 (July 23--25, 2014, Kobe, Japan
Timing sequence evolution, influencing factors and control approaches to China’s gross energy consumption
Confronted by the finite limits of its natural resources and damage to its ecosystem caused by its high level of energy consumption, China is taking the initiative in developing self-restrained. In early 2013, the Chinese government announced clear and specific quantitative targets for the control of gross energy consumption. This could bring gross energy consumption of coal equivalent to less than 4 billion tons by 2015. It would also demonstrate the Chinese government’s promotion of a viable ecological future. Despite the variable growth rate in China’s energy consumption in the past, over the last 30 years, the gross energy consumption has continued to grow exponentially and China now consumes more energy than any other nation. China’s energy consumption has been influenced most significantly by macro-economic growth, the development of industrial structure, residents’ consumption structure and urbanization. The mission to control gross energy consumption is going to be an arduous one. In order to achieve its targets, China should establish the allocation and implementation mechanism of control targets for gross energy consumption, implement carbon emission trading, promote policies on industrial transformation and low-carbon society construction, etc
- …