Approximation algorithms for optimization of real-valued general conjugate complex forms

Abstract

Complex polynomial optimization has recently gained more and more attention in both theory and practice. In this paper, we study the optimization of a real-valued general conjugate complex form over various popular constraint sets including the m-th roots of complex unity, the complex unit circle, and the complex unit sphere. A real-valued general conjugate complex form is a homogenous polynomial function of complex variables as well as their conjugates, and always takes real values. General conjugate form optimization is a wide class of complex polynomial optimization models, which include many homogenous polynomial optimization in the real domain with either discrete or continuous variables, and Hermitian quadratic form optimization as well as its higher degree extensions. All the problems under consideration are NP-hard in general and we focus on polynomial-time approximation algorithms with worst-case performance ratios. These approximation ratios improve previous results when restricting our problems to some special classes of complex polynomial optimization, and improve or equate previous results when restricting our problems to some special classes of polynomial optimization in the real domain. These algorithms are based on tensor relaxation and random sampling. Our novel technical contributions are to establish the first set of probability lower bounds for random sampling over the m-th root of unity, the complex unit circle, and the complex unit sphere, and propose the first polarization formula linking general conjugate forms and complex multilinear forms.

Conjugate Symmetric Data Transmission Control Method based on Machine Learning

“…Based on the above analysis, the conjugate symmetric data transmission path is optimized. In the actual data transmission process, the energy consumption between the same distance and different links will be different due to the influence of objective factors, so energy-based the path optimization method adds transmission consumption weights to each link, so that the network can better integrate with the objective environment in which it is located, thereby better improving the reliability of data transmission [11][12][13][14]. The transmission consumption weight matrix is as follows:…”

On decompositions and approximations of conjugate partial-symmetric complex tensors

“…, m with some finite m. This decomposition seems natural from its original definition, but is quite different to and less symmetric than (9) in Corollary 3.3. In fact, (10) can be immediately obtained from ( 9) by absorbing each λ j into a j ⊗d . This makes the decomposition (9) interesting as it links the first half and the last half modes of a PS tensor, which is not obvious either from Definition 2.1 or the decomposition (10).…”

“…In fact, (10) can be immediately obtained from ( 9) by absorbing each λ j into a j ⊗d . This makes the decomposition (9) interesting as it links the first half and the last half modes of a PS tensor, which is not obvious either from Definition 2.1 or the decomposition (10). Even for d = 1, (9) reduces to that any complex matrix A ∈ C n 2 can be written as A = m j=1 λ j a j a j H with λ j ∈ C and a j ∈ C n for j = 1, .…”

“…To echo the discussion at the end of Section 3.1, we remark that by the original definition (Definition 2.1) of PS tensors, another rank for PS tensors can be defined based on the decomposition (10), i.e., the minimum r such that T = r j=1 a j ⊗d ⊗ b j ⊗d . This rank is different to the PS rank in Definition 3.4 (see Example 3.6), and is not in the scope of this paper.…”

On decompositions and approximations of conjugate partial-symmetric tensors

“…For instance, every symmetric complex form generated by a CPS tensor is real-valued and all the eigenvalues of a CPS tensor are real [24]. In contrast to the many efforts on the optimization aspect [12,13,19,23,40,45] of CPS tensors, the current paper aims for their decompositions, ranks and approximations, which are important topics for high-order tensors. As we all know that the generalization of matrices to high-order tensors has led to interesting new findings as well as keeping many nice properties, CPS tensors, as a generalization of Hermitian matrices in terms of the high order and a generalization of real symmetric tensors in terms of the complex field, should also be expected to behave in that sense.…”