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The Copositive Cone, the Completely Positive Cone and their Generalisations
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... (2) Theoretical methods which can compute exact cp-factorizations in finitely many algorithmic steps. The factorization method of Anstreicher, Burer, and Dickinson [7,Section 3.3] uses the ellipsoid method and works for all matrices which have a rational cp-factorization and lie in the interior of the cone CP n . Berman and Rothblum [1] use quantifier elimination for first order formulae over the reals to compute the CP-rank of a given matrix, that is, the minimum number m of vectors used in a cp-factorization (1). ...
... As a consequence, to the best of our knowledge, our algorithm is currently the only one that can find a rational cp-factorization whenever it exists. In [7] a similar result was obtained, but restricted to matrices in the interior of CP n . A related question is if every rational completely positive matrix has a rational cp-factorization, see the survey [29]. ...
... A related question is if every rational completely positive matrix has a rational cp-factorization, see the survey [29]. Generally we do not know but from the results in [7] and [10] it follows that this is true for matrices in the interior of CP n . ...
In this paper we provide an algorithm, similar to the simplex algorithm, which determines a rational cp-factorization of a given matrix, whenever the matrix allows such a factorization. This algorithm can be used to show that every integral completely positive \(2 \times 2\) matrix has an integral cp-factorization.
... Several NP-hard problems can be formulated in this way. We provide a short list of some examples of such problems: (i) optimizing a homogeneous quadratic polynomial over the non-negative orthant, which is equivalent to testing the matrix copositivity [8,32]; (ii) 0-1 linear optimization problems [48,49,55]; (iii) quadratic optimization problems including, for example, the Quadratic assignment problem, the Graph partitioning problem and the MAX-CUT problem [10,28,40,41,42,46,45]. Polynomial optimization is also a strong tool in control, see [11]. ...
... • POP over the non-negative orthant, where we found that our hierarchy in its standard form performs poorly compared to state-of-the-art methods, but performs comparatively well when we added the redundant SDP constraint (8) to the original problem before applying our hierarchy. ...
... • PSDP over the non-negative orthant, where we found that our hierarchy (in its standard form) performs comparatively well in comparison to state-of-the-art methods, and performs even better when adding the redundant SDP constraint (8) to the original problem. ...
In this paper we consider polynomial conic optimization problems, where the feasible set is defined by constraints in the form of given polynomial vectors belonging to given nonempty closed convex cones, and we assume that all the feasible solutions are non-negative. This family of problems captures in particular polynomial optimization problems (POPs), polynomial semi-definite polynomial optimization problems (PSDPs) and polynomial second-order cone-optimization problems (PSOCPs). We propose a new general hierarchy of linear conic optimization relaxations inspired by an extension of Pólya’s Positivstellensatz for homogeneous polynomials being positive over a basic semi-algebraic cone contained in the non-negative orthant, introduced in Dickinson and Povh (J Glob Optim 61(4):615–625, 2015). We prove that based on some classic assumptions, these relaxations converge monotonically to the optimal value of the original problem. Adding a redundant polynomial positive semi-definite constraint to the original problem drastically improves the bounds produced by our method. We provide an extensive list of numerical examples that clearly indicate the advantages and disadvantages of our hierarchy. In particular, in comparison to the classic approach of sum-of-squares, our new method provides reasonable bounds on the optimal value for POPs, and strong bounds for PSDPs and PSOCPs, even outperforming the sum-of-squares approach in these latter two cases.
... A straightforward calculation shows that C R COP = COP n , hence by Lemma 7.5 we have to look at the facets of CP n . They were already investigated in [6]. Altogether we obtain the proof of the main result of this section: ...
... To end this section, we shortly discuss an application: In [10, Section 2.2] it was shown that the interior of COP n satisfies int COP n = cone R COP \ {0} (6) (which generalizes the classical case, where we have int S n ≥0 = cone R \ {0}). Hence Theorem 7.1 now gives us a kind of discretization of COP n in terms of the vertices and rays of R COP . ...
In this paper we give a first study of perfect copositive $n \times n$ matrices. They can be used to find rational certificates for completely positive matrices. We describe similarities and differences to classical perfect, positive definite matrices. Most of the differences occur only for $n \geq 3$, where we find for instance lower rank and indefinite perfect matrices. Nevertheless, we find for all $n$ that for every classical perfect matrix there is an arithmetically equivalent one which is also perfect copositive. Furthermore we study the neighborhood graph and polyhedral structure of perfect copositive matrices. As an application we obtain a new characterization of the cone of completely positive matrices: It is equal to the set of nonnegative matrices having a nonnegative inner product with all perfect copositive matrices.
... A straightforward calculation shows that C R COP = COP n , hence we have to look at the facets of CP n . They were already investigated in [6]. Altogether we obtain the Proof of Theorem 7.1. ...
... To end this section we want to shortly discuss an application of Theorem 7.1. In [11,Section 2.2] it was shown that the interior of COP n satisfies (6) int COP n = cone R COP \ {0} ...
In this paper we give a first study of perfect copositive $n \times n$ matrices. These can be used to find rational certificates for completely positive matrices. We describe similarities and differences to classical perfect, positive definite matrices. Most of the differences occur only for $n \geq 3$, where we find for instance lower rank and indefinite perfect matrices. Nevertheless we find in all $n$ that for every classical perfect matrix there is an arithmetically equivalent one which is also perfect copositive. Furthermore we study the neighborhood graph and polyhedral structure of perfect copositive matrices. As an application we obtain a new characterization of the cone of completely positive matrices: It is equal to the set of nonnegative matrices having a nonnegative inner product with all perfect copositive matrices.
... However, the standard reference, as far as linear algebra is concerned, is the classic book [13], which mostly deals with CPP(R n + ). Further developments on the analysis of COP(R n + ) and CPP(R n + ) can be found in [39,40,44], which present interesting geometrical and topological insights on the two cones. For many of these results it is still an open question, whether they can be generalized to cases where the ground cones differ from the non-negative orthant. ...
... The statement was proved first in [40] and then independently in [64], both for the case that K is closed. We present an alternative proof that merely requires o ∈ K. Sufficiency is clear, since any set-completely positive matrix cone is a subset of the positive semidefinite matrix cone and the north-west block of any matrix in CPP(K × R m ) is in CPP(K). ...
Robust optimization and stochastic optimization are the two main paradigms for dealing with the uncertainty inherent in almost all real-world optimization problems. The core principle of robust optimization is the introduction of parameterized families of constraints. Sometimes, these complicated semi-infinite constraints can be reduced to finitely many convex constraints, so that the resulting optimization problem can be solved using standard procedures. Hence flexibility of robust optimization is limited by certain convexity requirements on various objects. However, a recent strain of literature has sought to expand applicability of robust optimization by lifting variables to a properly chosen matrix space. Doing so allows to handle situations where convexity requirements are not met immediately, but rather intermediately.
In the domain of (possibly nonconvex) quadratic optimization, the principles of copositive optimization act as a bridge leading to recovery of the desired convex structures. Copositive optimization has established itself as a powerful paradigm for tackling a wide range of quadratically constrained quadratic optimization problems, reformulating them into linear convex-conic optimization problems involving only linear constraints and objective, plus constraints forcing membership to some matrix cones, which can be thought of as generalizations of the positive-semidefinite matrix cone. These reformulations enable application of powerful optimization techniques, most notably convex duality, to problems which, in their original form, are highly nonconvex.
In this text we want to offer readers an introduction and tutorial on these principles of copositive optimization, and to provide a review and outlook of the literature that applies these to optimization problems involving uncertainty.
... S n ++ is a cone and is the interior of S n + . It has been observed in [91] and in [52] that also the sets C n and C * n are closed, convex, full-dimensional pointed cones. All the mentioned cones share the same vertex O n . ...
... ,m. It is shown, for instance, in [52] (Section 1.2.2). Two main classes of linear conic optimization are well known, and they are defined based on the type of cones which are considered. ...
Non linear programming problems. There are several solution methods in literature for these problems, which are, however, not always efficient in general, in particular for large scale problems. Decomposition strategies such as Column Generation have been developed in order to substitute the original problem with a sequence of more tractable ones. One of the most known of these techniques is Dantzig-Wolfe Decomposition: it has been developed for linear problems and it consists in solving a sequence of subproblems, called respectively master and pricing programs, which leads to the optimum. This method can be extended to convex non linear problems and a classic example of this, which can be seen also as a generalization of the Frank-Wolfe algorithm, is Simplicial Decomposition(SD).In this thesis we discuss decomposition algorithms for solving quadratic optimization problems. In particular, we start with quadratic convex problems, both continuous and mixed binary. Then we tackle the more general class of binary quadratically constrained, quadratic problems. In the first part, we concentrate on SD based-methods for continuous, convex quadratic programming. We introduce new features in the algorithms, for both the master and the pricing problems of the decomposition, and provide results for a wide set of instances, showing that our algorithm is really efficient if compared to the state-of-the-art solver Cplex. This first work is accepted for publication in the journal Computational Optimization and Applications.We then extend the SD-based algorithm to mixed binary convex quadratic problems;we embed the continuous algorithm in a branch and bound scheme that makes us able to exploit some properties of our framework. In this context again we obtain results which show that in some sets of instances this algorithm is still more efficient than Cplex,even with a very simple branch and bound algorithm. This work is in preparation for submission to a journal. In the second part of the thesis, we deal with a more general class of problems, that is quadratically constrained, quadratic problems, where the constraints can be quadratic and both the objective function and the constraints can be non convex. For this class of problems we extend the formulation to the matrix space of the products of variables; we study an algorithm based on Dantzig-Wolfe Decomposition that exploits a relaxation on the Boolean Quadric Polytope (BQP), which is strictly contained in the Completely Positive cone and hence in the cone of positive semi definite (PSD) matrices. This is a constructive algorithm to solve the BQP relaxation of a binary problem an dwe obtain promising results for the root node bound for some quadratic problems. We compare our results with those obtained by the Semi definite relaxation of the ad-hocsolver BiqCrunch. We also show that, for linearly constrained quadratic problems, our relaxation can provide the integer optimum, under certain assumptions. We further study block decomposed matrices and provide results on the so-called BQP-completion problem ; these results are connected to those of PSD and CPP matrices. We show that, given a BQP matrix with some unspecified elements, it can be completed to a full BQP matrix under some assumptions on the positions of the specified elements. This result is related to optimization problems. We propose a BQP-relaxation based on the block structure of the problem. We prove that it provides a lower bound for the previously introduced relaxation, and that in some cases the two formulations are equivalent. We also conjecture that the equivalence result holds if and only if its so-called specification graph is chordal. We provide computational results which show the improvement in the performance of the block-based relaxation, with respect to the unstructured relaxation, and which support our conjecture. This work is in preparation for submission to a journal.
... In case K = R n the respective set-completely positive cone is simply the cone of positive semidefinite matrices. For more details on set-completely positive matrices see [3,11,20]. ...
... However, the fully specified submatrices given by show that the two sets have a common face and that the linear constraints in (3) confine the feasible set to that face, we have that the reformulation is actually tight. However, as discussed in [11], the identification of faces of CPP R n + is a difficult task in and of itself, so we cannot hope to easily obtain results on the faces of CPP S,k n × S i=0 K i and CMP S,k n × S i=0 K i . We leave such a discussion to future research and close of this article with a discussion on the consequences that Theorem 3 has for matrix completion. ...
In recent literature, many successful attempts have been made in reformulating non-convex optimization problems as convex conic optimization problems, or at least in giving powerful relaxations of this type. However, it is often the case that these approaches suffer from poor computational performance due to the size of the reformulation. In this paper we study QCQPs whose special structure allows for a reduced conic relaxation where the variables are lifted into a smaller space compared to traditional approaches. We show that in special cases these relaxations are exact reformulations. We contextualized these results in the field of matrix completion in two ways. First we derive an exact reformulation of the QCQPs that is based on the geometry of the cone of partial arrowhead matrix that have a set-completely positive completion. However, we do not reference the completability in the proof, but instead rely purely on geometric aspects. We also show, that our reduced conic reformulation can be seen as relaxation of the latter exact reformulation and describe the gap between those two in terms of the difference of two subsets of a space that is isomorphic to the space of arrowhead matrices if a certain type. Second we show that our partial exactness results imply a new sufficient conditions for set-completely positive matrix completion.
... The copositive and completely positive cones of matrices, which are tensors of order two, are very well explored (see, e.g., [10][11][12] and also [13] for a list of open problems). Therefore, it seems natural to study similar results for copositive tensors. ...
In this paper, we discuss the cone of copositive tensors and its approximation. We describe some basic properties of copositive tensors and positive semidefinite tensors. Specifically, we show that a non-positive tensor (or Z-tensor) is copositive if and only if it is positive semidefinite. We also describe cone hierarchies that approximate the copositive cone. These hierarchies are based on the sum of squares conditions and the non-negativity of polynomial coefficients. We provide a compact representation for the approximation based on the non-negativity of polynomial coefficients. As an immediate consequence of this representation, we show that the approximation based on the non-negativity of polynomial coefficients is polyhedral. Furthermore, these hierarchies are used to provide approximation results for optimizing a (homogeneous) polynomial over the simplex.
... again with a large-order psd constraint. Inspired by Burer (2010) and Dickinson (2013), another reformulation of (2) was recently put forward in Bomze et al. (2017) which combines both above approaches (and performance advantages): ...
By now, many copositive reformulations of mixed-binary QPs have been discussed, triggered by Burer’s seminal characterization from 2009. In conic optimization, it is very common to use approximation hierarchies based on positive-semidefinite (psd) matrices where the order increases with the level of the approximation. Our purpose is to keep the psd matrix orders relatively small to avoid memory size problems in interior point solvers. Based upon on a recent discussion on various variants of completely positive reformulations and their relaxations (Bomze et al. in Math Program 166(1–2):159–184, 2017), we here present a small study of the notoriously hard multidimensional quadratic knapsack problem and quadratic assignment problem. Our observations add some empirical evidence on performance differences among the above mentioned variants. We also propose an alternative approach using penalization of various classes of (aggregated) constraints, along with some theoretical convergence analysis. This approach is in some sense similar in spirit to the alternating projection method proposed in Burer (Math Program Comput 2:1–19, 2010) which completely avoids SDPs, but for which no convergence proof is available yet.
In the present article, we discuss a general concept of lifting a non-convex quadratic optimization problem into a convex program with matrix variables, and then apply it to construct two kinds of equivalent lifted problems, which are completely positive programs (linear programs with completely positive matrix variables) for a class of quadratic optimization problem with linear inequality and mixed binary constraints. The duals of the resulting completely positive programs, which are copositive programs, are constructed. It is shown that, under some conditions, the dual problems are strictly feasible, such that strong duality holds and existing numerical methods for both primal and dual problems can be applied.
In this paper we consider polynomial conic optimization problems, where the feasible set is defined by constraints in the form of given polynomial vectors belonging to given nonempty closed convex cones, and we assume that all the feasible solutions are non-negative. This family of problems captures in particular polynomial optimization problems (POPs), polynomial semi-definite polynomial optimization problems (PSDPs) and polynomial second-order cone-optimization problems (PSOCPs). We propose a new general hierarchy of linear conic optimization relaxations inspired by an extension of Pólya’s Positivstellensatz for homogeneous polynomials being positive over a basic semi-algebraic cone contained in the non-negative orthant, introduced in Dickinson and Povh (J Glob Optim 61(4):615–625, 2015). We prove that based on some classic assumptions, these relaxations converge monotonically to the optimal value of the original problem. Adding a redundant polynomial positive semi-definite constraint to the original problem drastically improves the bounds produced by our method. We provide an extensive list of numerical examples that clearly indicate the advantages and disadvantages of our hierarchy. In particular, in comparison to the classic approach of sum-of-squares, our new method provides reasonable bounds on the optimal value for POPs, and strong bounds for PSDPs and PSOCPs, even outperforming the sum-of-squares approach in these latter two cases.
In this paper set-semidefinite optimization is introduced as a new field of vector optimization in infinite dimensions covering semidefinite and copositive programming. This unified approach is based on a special ordering cone, the so-called K-semidefinite cone for which properties are given in detail. Optimality conditions in the KKT form and duality results including the linear case are presented for K-semidefinite optimization problems. A penalty approach is developed for the treatment of the special constraint arising in K-semidefinite optimization problems.
A real symmetric matrix A is said to be copositive if (Ax, x) ≥ 0 for every vector x with nonnegative components. In a recent publication, Bomze proposed a linear-time algorithm for verifying the copositivity of a tridiagonal matrix. It is shown that this algorithm can be extended to the case of an arbitrary acyclic matrix.
