In this article, we present various new results on Cauchy tensors and Hankel tensors. We first in... more In this article, we present various new results on Cauchy tensors and Hankel tensors. We first introduce the concept of generalized Cauchy tensors which extends Cauchy tensors in the current literature, and provide several conditions characterizing positive semi-definiteness of generalized Cauchy tensors with nonzero entries. As a consequence, we show that Cauchy tensors are positive semidefinite if and only if they are SOS (Sum-of-squares) tensors. Furthermore, we prove that all positive semi-definite Cauchy tensors are completely positive tensors, which means every positive semi-definite Cauchy tensor can be decomposed as the sum of nonnegative rank-1 tensors. We also establish that all the H-eigenvalues of nonnegative Cauchy tensors are nonnegative. Secondly, we present new mathematical properties of Hankel tensors. We prove that an even order Hankel tensor is Vandermonde positive semi-definite if and only if its associated plane tensor is positive semi-definite. We also show that, if the Vandermonde rank of a Hankel tensor A is less than the dimension of the underlying space, then positive semi-definiteness of A is equivalent to the fact that A is a complete Hankel tensor, and so, is further equivalent to the SOS property of A. Lastly, we introduce a new structured tensor called Cauchy-Hankel tensors, which is a special case of Cauchy tensors and Hankel tensors simultaneously. Sufficient and necessary conditions are established for an even order Cauchy-Hankel tensor to be positive definite. Final remarks are listed at the end of the paper.
ABSTRACT Are there positive semi-definite (PSD) non-SOS Hankel tensors? If the answer to this que... more ABSTRACT Are there positive semi-definite (PSD) non-SOS Hankel tensors? If the answer to this question is no, then the problem for determining an even order Hankel tensor is PSD or not is solvable in polynomial-time. By Hilbert, one of the cases of low order (degree) and dimension (number of variables), in which there are PSD non-SOS (PNS) symmetric tensors (homogeneous polynomials), is of order six and dimension three. The famous Motzkin polynomial is of degree six with three variables. In this paper, we study the existence problem of sixth order three dimensional PNS Hankel tensors. We study several special cases of sixth order three dimensional Hankel tensors. No PNS Hankel tensors are found in these cases. We then randomly generate several thousands of sixth order three dimensional Hankel tensors and make them PSD by adding adequate multiple of a fixed sixth order three dimensional positive definite Hankel tensor. Again, still no PNS Hankel tensors are found. Thus, we make a conjecture that there are no sixth order three dimensional PNS Hankel tensors. This implies that the problem for determining a given sixth order three dimensional Hankel tensor is PSD or not can be solved by a semi-definite linear programming problem.
In this article, we present various new results on Cauchy tensors and Hankel tensors. We first in... more In this article, we present various new results on Cauchy tensors and Hankel tensors. We first introduce the concept of generalized Cauchy tensors which extends Cauchy tensors in the current literature, and provide several conditions characterizing positive semi-definiteness of generalized Cauchy tensors with nonzero entries. As a consequence, we show that Cauchy tensors are positive semidefinite if and only if they are SOS (Sum-of-squares) tensors. Furthermore, we prove that all positive semi-definite Cauchy tensors are completely positive tensors, which means every positive semi-definite Cauchy tensor can be decomposed as the sum of nonnegative rank-1 tensors. We also establish that all the H-eigenvalues of nonnegative Cauchy tensors are nonnegative. Secondly, we present new mathematical properties of Hankel tensors. We prove that an even order Hankel tensor is Vandermonde positive semi-definite if and only if its associated plane tensor is positive semi-definite. We also show that, if the Vandermonde rank of a Hankel tensor A is less than the dimension of the underlying space, then positive semi-definiteness of A is equivalent to the fact that A is a complete Hankel tensor, and so, is further equivalent to the SOS property of A. Lastly, we introduce a new structured tensor called Cauchy-Hankel tensors, which is a special case of Cauchy tensors and Hankel tensors simultaneously. Sufficient and necessary conditions are established for an even order Cauchy-Hankel tensor to be positive definite. Final remarks are listed at the end of the paper.
ABSTRACT Anti-circulant tensors have applications in exponential data fitting. They are special H... more ABSTRACT Anti-circulant tensors have applications in exponential data fitting. They are special Hankel tensors. In this paper, we extend the definition of anti-circulant tensors to generalized anti-circulant tensors by introducing a circulant index $r$ such that the entries of the generating vector of a Hankel tensor is circulant with module $r$. For an anti-circulant tensor, r=n, where n is the dimension of the Hankel tensor. For generalized anti-circulant tensors, we allow r <= n. Hence, generalized anti-circulant tensors are still special Hankel tensors. For r <= 4, we give necessary and sufficient conditions for positive semi-definiteness of even order generalized anti-circulant tensors, and show that in these cases, they are SOS tensors. This shows that there are no PNS Hankel tensors in these cases. For r => 5, we show that these results are still true in some cases.
The multiobjective optimization model studied in this paper deals with simultaneous minimization ... more The multiobjective optimization model studied in this paper deals with simultaneous minimization of finitely many linear functions subject to an arbitrary number of uncertain linear constraints. We first provide a radius of robust feasibility guaranteeing the feasibility of the robust counterpart under affine data parametrization. We then establish dual characterizations of robust solutions of our model that are immunized against data uncertainty by way of characterizing corresponding solutions of robust counterpart of the model. Consequently, we present robust duality theorems relating the value of the robust model with the corresponding value of its dual problem.
In this paper, we present some new and tractable sufficient conditions for convex asymptotic well... more In this paper, we present some new and tractable sufficient conditions for convex asymptotic well behaved (AWB) functions. Moreover, we establish several Lipschitz/Hölder type global error bound results for single convex polynomial and for function which can be expressed as maximum of finitely many nonnegative convex polynomials. An advantage of our approach is that the corresponding Hölder exponent in our Hölder type global error bound results can be determined explicitly. .au † Sometimes, we also use x R m (in stead of x ), if the underlying space need to be specified
We consider the problem of minimizing the sum of a smooth function $h$ with a bounded Hessian, an... more We consider the problem of minimizing the sum of a smooth function $h$ with a bounded Hessian, and a nonsmooth function. We assume that the latter function is a composition of a proper closed function $P$ and a surjective linear map $\cal M$, with the proximal mappings of $\tau P$, $\tau > 0$, simple to compute. This problem is nonconvex in general and encompasses many important applications in engineering and machine learning. In this paper, we examined two types of splitting methods for solving this nonconvex optimization problem: alternating direction method of multipliers and proximal gradient algorithm. For the direct adaptation of the alternating direction method of multipliers, we show that, if the penalty parameter is chosen sufficiently large and the sequence generated has a cluster point, then it gives a stationary point of the nonconvex problem. We also establish convergence of the whole sequence under an additional assumption that the functions $h$ and $P$ are semi-algebraic. Furthermore, we give simple sufficient conditions to guarantee boundedness of the sequence generated. These conditions can be satisfied for a wide range of applications including the least squares problem with the $\ell_{1/2}$ regularization. Finally, when $\cal M$ is the identity so that the proximal gradient algorithm can be efficiently applied, we show that any cluster point is stationary under a slightly more flexible constant step-size rule than what is known in the literature for a nonconvex $h$.
This paper addresses the issue of which nonlinear semidefinite linear programming problems posses... more This paper addresses the issue of which nonlinear semidefinite linear programming problems possess exact semidefinite linear programming (SDP) relaxations under a constraint qualification. We establish exact SDP relaxations for classes of nonlinear semidefinite programming problems with SOS-convex polynomials. These classes include SOS-convex semidefinite programming problems and fractional semidefinite programming problems with SOS-convex polynomials. The class of SOS-convex polynomials contains convex quadratic functions and separable convex polynomials. We also derive numerically checkable conditions, completely characterizing minimizers of these classes of problems.
We are glad to achieve this special issue. This special issue was opened in late November of 2013... more We are glad to achieve this special issue. This special issue was opened in late November of 2013 and closed in early February of 2014. There were 67 submissions in total, and 24 of them were accepted for publication after strict reviews, which gave important developments and applications in nonlinear analysis.
In this article, we present various new results on Cauchy tensors and Hankel tensors. We first in... more In this article, we present various new results on Cauchy tensors and Hankel tensors. We first introduce the concept of generalized Cauchy tensors which extends Cauchy tensors in the current literature, and provide several conditions characterizing positive semi-definiteness of generalized Cauchy tensors with nonzero entries. As a consequence, we show that Cauchy tensors are positive semidefinite if and only if they are SOS (Sum-of-squares) tensors. Furthermore, we prove that all positive semi-definite Cauchy tensors are completely positive tensors, which means every positive semi-definite Cauchy tensor can be decomposed as the sum of nonnegative rank-1 tensors. We also establish that all the H-eigenvalues of nonnegative Cauchy tensors are nonnegative. Secondly, we present new mathematical properties of Hankel tensors. We prove that an even order Hankel tensor is Vandermonde positive semi-definite if and only if its associated plane tensor is positive semi-definite. We also show that, if the Vandermonde rank of a Hankel tensor A is less than the dimension of the underlying space, then positive semi-definiteness of A is equivalent to the fact that A is a complete Hankel tensor, and so, is further equivalent to the SOS property of A. Lastly, we introduce a new structured tensor called Cauchy-Hankel tensors, which is a special case of Cauchy tensors and Hankel tensors simultaneously. Sufficient and necessary conditions are established for an even order Cauchy-Hankel tensor to be positive definite. Final remarks are listed at the end of the paper.
ABSTRACT Are there positive semi-definite (PSD) non-SOS Hankel tensors? If the answer to this que... more ABSTRACT Are there positive semi-definite (PSD) non-SOS Hankel tensors? If the answer to this question is no, then the problem for determining an even order Hankel tensor is PSD or not is solvable in polynomial-time. By Hilbert, one of the cases of low order (degree) and dimension (number of variables), in which there are PSD non-SOS (PNS) symmetric tensors (homogeneous polynomials), is of order six and dimension three. The famous Motzkin polynomial is of degree six with three variables. In this paper, we study the existence problem of sixth order three dimensional PNS Hankel tensors. We study several special cases of sixth order three dimensional Hankel tensors. No PNS Hankel tensors are found in these cases. We then randomly generate several thousands of sixth order three dimensional Hankel tensors and make them PSD by adding adequate multiple of a fixed sixth order three dimensional positive definite Hankel tensor. Again, still no PNS Hankel tensors are found. Thus, we make a conjecture that there are no sixth order three dimensional PNS Hankel tensors. This implies that the problem for determining a given sixth order three dimensional Hankel tensor is PSD or not can be solved by a semi-definite linear programming problem.
In this article, we present various new results on Cauchy tensors and Hankel tensors. We first in... more In this article, we present various new results on Cauchy tensors and Hankel tensors. We first introduce the concept of generalized Cauchy tensors which extends Cauchy tensors in the current literature, and provide several conditions characterizing positive semi-definiteness of generalized Cauchy tensors with nonzero entries. As a consequence, we show that Cauchy tensors are positive semidefinite if and only if they are SOS (Sum-of-squares) tensors. Furthermore, we prove that all positive semi-definite Cauchy tensors are completely positive tensors, which means every positive semi-definite Cauchy tensor can be decomposed as the sum of nonnegative rank-1 tensors. We also establish that all the H-eigenvalues of nonnegative Cauchy tensors are nonnegative. Secondly, we present new mathematical properties of Hankel tensors. We prove that an even order Hankel tensor is Vandermonde positive semi-definite if and only if its associated plane tensor is positive semi-definite. We also show that, if the Vandermonde rank of a Hankel tensor A is less than the dimension of the underlying space, then positive semi-definiteness of A is equivalent to the fact that A is a complete Hankel tensor, and so, is further equivalent to the SOS property of A. Lastly, we introduce a new structured tensor called Cauchy-Hankel tensors, which is a special case of Cauchy tensors and Hankel tensors simultaneously. Sufficient and necessary conditions are established for an even order Cauchy-Hankel tensor to be positive definite. Final remarks are listed at the end of the paper.
ABSTRACT Anti-circulant tensors have applications in exponential data fitting. They are special H... more ABSTRACT Anti-circulant tensors have applications in exponential data fitting. They are special Hankel tensors. In this paper, we extend the definition of anti-circulant tensors to generalized anti-circulant tensors by introducing a circulant index $r$ such that the entries of the generating vector of a Hankel tensor is circulant with module $r$. For an anti-circulant tensor, r=n, where n is the dimension of the Hankel tensor. For generalized anti-circulant tensors, we allow r <= n. Hence, generalized anti-circulant tensors are still special Hankel tensors. For r <= 4, we give necessary and sufficient conditions for positive semi-definiteness of even order generalized anti-circulant tensors, and show that in these cases, they are SOS tensors. This shows that there are no PNS Hankel tensors in these cases. For r => 5, we show that these results are still true in some cases.
The multiobjective optimization model studied in this paper deals with simultaneous minimization ... more The multiobjective optimization model studied in this paper deals with simultaneous minimization of finitely many linear functions subject to an arbitrary number of uncertain linear constraints. We first provide a radius of robust feasibility guaranteeing the feasibility of the robust counterpart under affine data parametrization. We then establish dual characterizations of robust solutions of our model that are immunized against data uncertainty by way of characterizing corresponding solutions of robust counterpart of the model. Consequently, we present robust duality theorems relating the value of the robust model with the corresponding value of its dual problem.
In this paper, we present some new and tractable sufficient conditions for convex asymptotic well... more In this paper, we present some new and tractable sufficient conditions for convex asymptotic well behaved (AWB) functions. Moreover, we establish several Lipschitz/Hölder type global error bound results for single convex polynomial and for function which can be expressed as maximum of finitely many nonnegative convex polynomials. An advantage of our approach is that the corresponding Hölder exponent in our Hölder type global error bound results can be determined explicitly. .au † Sometimes, we also use x R m (in stead of x ), if the underlying space need to be specified
We consider the problem of minimizing the sum of a smooth function $h$ with a bounded Hessian, an... more We consider the problem of minimizing the sum of a smooth function $h$ with a bounded Hessian, and a nonsmooth function. We assume that the latter function is a composition of a proper closed function $P$ and a surjective linear map $\cal M$, with the proximal mappings of $\tau P$, $\tau > 0$, simple to compute. This problem is nonconvex in general and encompasses many important applications in engineering and machine learning. In this paper, we examined two types of splitting methods for solving this nonconvex optimization problem: alternating direction method of multipliers and proximal gradient algorithm. For the direct adaptation of the alternating direction method of multipliers, we show that, if the penalty parameter is chosen sufficiently large and the sequence generated has a cluster point, then it gives a stationary point of the nonconvex problem. We also establish convergence of the whole sequence under an additional assumption that the functions $h$ and $P$ are semi-algebraic. Furthermore, we give simple sufficient conditions to guarantee boundedness of the sequence generated. These conditions can be satisfied for a wide range of applications including the least squares problem with the $\ell_{1/2}$ regularization. Finally, when $\cal M$ is the identity so that the proximal gradient algorithm can be efficiently applied, we show that any cluster point is stationary under a slightly more flexible constant step-size rule than what is known in the literature for a nonconvex $h$.
This paper addresses the issue of which nonlinear semidefinite linear programming problems posses... more This paper addresses the issue of which nonlinear semidefinite linear programming problems possess exact semidefinite linear programming (SDP) relaxations under a constraint qualification. We establish exact SDP relaxations for classes of nonlinear semidefinite programming problems with SOS-convex polynomials. These classes include SOS-convex semidefinite programming problems and fractional semidefinite programming problems with SOS-convex polynomials. The class of SOS-convex polynomials contains convex quadratic functions and separable convex polynomials. We also derive numerically checkable conditions, completely characterizing minimizers of these classes of problems.
We are glad to achieve this special issue. This special issue was opened in late November of 2013... more We are glad to achieve this special issue. This special issue was opened in late November of 2013 and closed in early February of 2014. There were 67 submissions in total, and 24 of them were accepted for publication after strict reviews, which gave important developments and applications in nonlinear analysis.
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