Nonnegative Orthant Research Articles

Positive functional filters synthesis for positive linear systems

Abstract This paper deals with the problem of positive functional $H_{\infty }$ filtering for positive linear systems subject to both bounded disturbances and unknown inputs. The order of the designed $H_{\infty }$ filter is equal to the dimension of the functional to be estimated, that can be useful for control purpose for example. We focus on the design of positive filters which guarantee that the functional (of the state) is always non-negative at any time and converges to the real functional state by minimizing the effect of the disturbances on the estimation error. In fact, we propose a new positive reduced order $H_{\infty }$ filter for positive linear systems for which the states remain in the non-negative orthant of the state space, subject to both disturbances and unknown inputs. This paper is a first attempt to design positive unknown inputs functional $H_{\infty }$ filter for such positive linear systems. The proposed approach is based on the positivity of an augmented system composed of dynamics of both considered system and proposed filter, on the unbiasedness of the estimation error by the resolution of Sylvester equation; then we derive conditions for the establishment of such filters in terms of an optimization problem. An algorithm that summarizes the different steps of the proposed positive functional $H_{\infty }$ filter design is given. Finally, numerical example and simulation results are given to showcase the effectiveness of the proposed design method.

Closest Farthest Widest

The current paper proposes and tests algorithms for finding the diameter of a compact convex set and the farthest point in the set to another point. For these two nonconvex problems, I construct Frank–Wolfe and projected gradient ascent algorithms. Although these algorithms are guaranteed to go uphill, they can become trapped by local maxima. To avoid this defect, I investigate a homotopy method that gradually deforms a ball into the target set. Motivated by the Frank–Wolfe algorithm, I also find the support function of the intersection of a convex cone and a ball centered at the origin and elaborate a known bisection algorithm for calculating the support function of a convex sublevel set. The Frank–Wolfe and projected gradient algorithms are tested on five compact convex sets: (a) the box whose coordinates range between −1 and 1, (b) the intersection of the unit ball and the non-negative orthant, (c) the probability simplex, (d) the Manhattan-norm unit ball, and (e) a sublevel set of the elastic net penalty. Frank–Wolfe and projected gradient ascent are about equally fast on these test problems. Ignoring homotopy, the Frank–Wolfe algorithm is more reliable. However, homotopy allows projected gradient ascent to recover from its failures.

Lorentzian polynomials, Segre classes, and adjoint polynomials of convex polyhedral cones

We consider polynomials expressing the cohomology classes of subvarieties of products of projective spaces, and limits of positive real multiples of such polynomials. We study the relation between these covolume polynomials and Lorentzian polynomials. While these are distinct notions, we prove that, like Lorentzian polynomials, covolume polynomials have M-convex support and generalize the notion of log-concave sequences. In fact, we prove that covolume polynomials are ‘sectional log-concave’, that is, the coefficients of suitable restrictions of these polynomials form log-concave sequences.We observe that Chern classes of globally generated bundles give rise to covolume polynomials, and use this fact to prove that certain polynomials associated with Segre classes of subschemes of products of projective spaces are covolume polynomials. We conjecture that the same polynomials may be Lorentzian after a standard normalization operation.Finally, we obtain a combinatorial application of a particular case of our Segre class result. We prove that the adjoint polynomial of a convex polyhedral cone contained in the nonnegative orthant, and sharing a face with it, is a covolume polynomials. This implies that these adjoint polynomials are M-convex and sectional log-concave, and in fact dually Lorentzian, that is, Lorentzian after a certain change of variables.

Addressing Machine Learning Problems in the Non-Negative Orthant

Addressing Machine Learning Problems in the Non-Negative Orthant

Computation of the Hausdorff Distance between Two Compact Convex Sets.

The Hausdorff distance between two closed sets has important theoretical and practical applications. Yet apart from finite point clouds, there appear to be no generic algorithms for computing this quantity. Because many infinite sets are defined by algebraic equalities and inequalities, this a huge gap. The current paper constructs Frank-Wolfe and projected gradient ascent algorithms for computing the Hausdorff distance between two compact convex sets. Although these algorithms are guaranteed to go uphill, they can become trapped by local maxima. To avoid this defect, we investigate a homotopy method that gradually deforms two balls into the two target sets. The Frank-Wolfe and projected gradient algorithms are tested on two pairs of compact convex sets, where: (1) is the box translated by and is the intersection of the unit ball and the non-negative orthant; and (2) is the probability simplex and is the unit ball translated by . For problem (2), we find the Hausdorff distance analytically. Projected gradient ascent is more reliable than the Frank-Wolfe algorithm and finds the exact solution of problem (2). Homotopy improves the performance of both algorithms when the exact solution is unknown or unattained.

A new design of positive functional filters for positive linear time-delay systems

This paper solves the problem of designing positive functional filters for positive linear time-delay systems. In fact, we propose a new positive reduced order filter for positive linear constant state time-delay systems for which the states remain in the nonnegative orthant of the state space, subject to disturbances and unknown inputs. This paper is a first attempt to design positive unknown inputs functional filter for such positive linear delayed systems. The proposed approach is based on the positivity of an augmented system composed of dynamics of both considered system and proposed filter, on the unbiasedness of the estimation error by the resolution of Sylvester equation and also on Lyapunov-Krasovskii stability theory; then conditions for the establishment of such filters are formulated in terms of an optimization problem. An algorithm that summarizes the different steps of the proposed positive functional filter design is given. Finally, numerical example and simulation results are given to illustrate the effectiveness of the proposed design method.

Approximation hierarchies for copositive cone over symmetric cone and their comparison

We first provide an inner-approximation hierarchy described by a sum-of-squares (SOS) constraint for the copositive (COP) cone over a general symmetric cone. The hierarchy is a generalization of that proposed by Parrilo (Structured semidefinite programs and semialgebraic geometry methods in Robustness and optimization, Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 2000) for the usual COP cone (over a nonnegative orthant). We also discuss its dual. Second, we characterize the COP cone over a symmetric cone using the usual COP cone. By replacing the usual COP cone appearing in this characterization with the inner- or outer-approximation hierarchy provided by de Klerk and Pasechnik (SIAM J Optim 12(4):875–892, https://doi.org/10.1137/S1052623401383248, 2002) or Yıldırım (Optim Methods Softw 27(1):155–173, https://doi.org/10.1080/10556788.2010.540014, 2012), we obtain an inner- or outer-approximation hierarchy described by semidefinite but not by SOS constraints for the COP matrix cone over the direct product of a nonnegative orthant and a second-order cone. We then compare them with the existing hierarchies provided by Zuluaga et al. (SIAM J Optim 16(4):1076–1091, https://doi.org/10.1137/03060151X, 2006) and Lasserre (Math Program 144:265–276, https://doi.org/10.1007/s10107-013-0632-5, 2014). Theoretical and numerical examinations imply that we can numerically increase a depth parameter, which determines an approximation accuracy, in the approximation hierarchies derived from de Klerk and Pasechnik (SIAM J Optim 12(4):875–892, https://doi.org/10.1137/S1052623401383248, 2002) and Yıldırım (Optim Methods Softw 27(1):155–173, https://doi.org/10.1080/10556788.2010.540014, 2012), particularly when the nonnegative orthant is small. In such a case, the approximation hierarchy derived from Yıldırım (Optim Methods Softw 27(1):155–173, https://doi.org/10.1080/10556788.2010.540014, 2012) can yield nearly optimal values numerically. Combining the proposed approximation hierarchies with existing ones, we can evaluate the optimal value of COP programming problems more accurately and efficiently.

Heavy-traffic limits for parallel single-server queues with randomly split Hawkes arrival processes

AbstractWe consider parallel single-server queues in heavy traffic with randomly split Hawkes arrival processes. The service times are assumed to be independent and identically distributed (i.i.d.) in each queue and are independent in different queues. In the critically loaded regime at each queue, it is shown that the diffusion-scaled queueing and workload processes converge to a multidimensional reflected Brownian motion in the non-negative orthant with orthonormal reflections. For the model with abandonment, we also show that the corresponding limit is a multidimensional reflected Ornstein–Uhlenbeck diffusion in the non-negative orthant.

A new extension of Chubanov’s method to symmetric cones

We propose a new variant of Chubanov’s method for solving the feasibility problem over the symmetric cone by extending Roos’s method (Optim Methods Softw 33(1):26–44, 2018) of solving the feasibility problem over the nonnegative orthant. The proposed method considers a feasibility problem associated with a norm induced by the maximum eigenvalue of an element and uses a rescaling focusing on the upper bound for the sum of eigenvalues of any feasible solution to the problem. Its computational bound is (1) equivalent to that of Roos’s original method (2018) and superior to that of Lourenço et al.’s method (Math Program 173(1–2):117–149, 2019) when the symmetric cone is the nonnegative orthant, (2) superior to that of Lourenço et al.’s method (2019) when the symmetric cone is a Cartesian product of second-order cones, (3) equivalent to that of Lourenço et al.’s method (2019) when the symmetric cone is the simple positive semidefinite cone, and (4) superior to that of Pena and Soheili’s method (Math Program 166(1–2):87–111, 2017) for any simple symmetric cones under the feasibility assumption of the problem imposed in Pena and Soheili’s method (2017). We also conduct numerical experiments that compare the performance of our method with existing methods by generating strongly (but ill-conditioned) feasible instances. For any of these instances, the proposed method is rather more efficient than the existing methods in terms of accuracy and execution time.

Combinatorics and preservation of conically stable polynomials

Given a closed, convex cone Ksubseteq mathbb {R}^n, a multivariate polynomial fin mathbb {C}[textbf{z}] is called K-stable if the imaginary parts of its roots are not contained in the relative interior of K. If K is the nonnegative orthant, K-stability specializes to the usual notion of stability of polynomials. We develop generalizations of preservation operations and of combinatorial criteria from usual stability toward conic stability. A particular focus is on the cone of positive semidefinite matrices (psd-stability). In particular, we prove the preservation of psd-stability under a natural generalization of the inversion operator. Moreover, we give conditions on the support of psd-stable polynomials and characterize the support of special families of psd-stable polynomials.

Automorphisms of Rank-One Generated Hyperbolicity Cones and Their Derivative Relaxations

A hyperbolicity cone is said to be rank-one generated (ROG) if all its extreme rays have rank 1, where the rank is computed with respect to the underlying hyperbolic polynomial. This is a natural class of hyperbolicity cones which are strictly more general than the ROG spectrahedral cones. In this work, we present a study of the automorphisms of ROG hyperbolicity cones and their derivative relaxations. One of our main results states that the automorphisms of the derivative relaxations are exactly the automorphisms of the original cone fixing a certain direction. As an application, we completely determine the automorphisms of the derivative relaxations of the nonnegative orthant and of the cone of positive semidefinite matrices. More generally, we also prove relations between the automorphisms of a spectral cone and the underlying permutation-invariant set, which might be of independent interest.

On eigenvalues of symmetric matrices with PSD principal submatrices

We investigate convexity properties of the set of eigenvalue tuples of n×n real symmetric matrices, all of whose k×k (where k≤n is fixed) minors are positive semidefinite. We prove that the set λ(Sn,k) of eigenvalue vectors of all such matrices is star-shaped with respect to the nonnegative orthant R≥0n and not convex already when (n,k)=(4,2). We also show that k is the smallest integer such that Sn,k is a linear projection of a set described by linear matrix inequalities of size k.

Cancer cell eradication in a 6D metastatic tumor model with time delay

In this paper we study ultimate dynamics of one time delay cancer metastatic 6D model with chemotherapy. This model describes interactions between normal and cancer cells under chemotherapy agents that occurs in two tumor sites, primary and secondary. We derive ultimate upper bounds for normal and cancer cells; ultimate upper/lower bounds for the chemical concentration in the both of tumor sites and describe the location of the attracting set. The study of equilibrium points is included. Global asymptotic tumor eradication conditions are obtained by a two-step process. Firstly, using the localization method of compact invariant sets (LMCIS) we find conditions of the tumor clearance at the primary site. Further, exploiting the cascade structure of the whole system and properties of asymptotically autonomous and competitive systems we derive tumor eradication conditions at the secondary site under the assumption that cancer cells at the primary site are asymptotically eradicated. Based on these results, cancer eradication conditions of the whole system are obtained. In particular, it is shown that the ω-limit set of each trajectory starting in the nonnegative orthant is one of equilibrium points located in the plane which is free from both primary and secondary cancer cells.

Low-rank nonnegative tensor approximation via alternating projections and sketching

We show how to construct nonnegative low-rank approximations of nonnegative tensors in Tucker and tensor train formats. We use alternating projections between the nonnegative orthant and the set of low-rank tensors, using STHOSVD and TTSVD algorithms, respectively, and further accelerate the alternating projections using randomized sketching. The numerical experiments on both synthetic data and hyperspectral images show the decay of the negative elements and that the error of the resulting approximation is close to the initial error obtained with STHOSVD and TTSVD. The proposed method for the Tucker case is superior to the previous ones in terms of computational complexity and decay of negative elements. The tensor train case, to the best of our knowledge, has not been studied before.

Almost sure convergence of randomized urn models with application to elephant random walk

We consider a randomized urn model with objects of finitely many colors. The replacement matrices are random, and are conditionally independent of the color chosen given the past. Further, the conditional expectations of the replacement matrices are close to an almost surely irreducible matrix. We obtain almost sure and L1 convergence of the configuration vector, the proportion vector and the count vector. We show that first moment is sufficient for i.i.d. replacement matrices independent of past color choices. This significantly improves the similar results for urn models obtained in Athreya and Ney (1972) requiring Llog+L moments. For more general adaptive sequence of replacement matrices, a little more than Llog+L condition is required. Similar results based on L1 moment assumption alone has been considered independently and in parallel in Zhang (2018). Finally, using the result, we study a delayed elephant random walk on the nonnegative orthant in d dimension with random memory.

Accelerating the gradient projection algorithm for solving the non-additive traffic equilibrium problem with the Barzilai-Borwein step size

The non-additive traffic equilibrium problem (NaTEP) overcomes the inadequacies of the additivity assumption in traditional traffic equilibrium models by relaxing the cost incurred on each path that is not a simple sum of the link costs on that path. The computation of the NaTEP heavily depends on the efficiency of the step size determination. This paper aims to accelerate the path-based gradient projection (GP) algorithm for solving the NaTEP using the Barzilai-Borwein (BB) step size scheme. The GP algorithm with the BB step size scheme uses the solution information of the last two iterations to determine a suitable step size and avoids extra evaluations of the mapping value. The proposed algorithm only needs to perform one-time projection onto the nonnegative orthant at each iteration. Two approaches with and without column generation are considered in the GP algorithm implementation. A non-additive shortest path algorithm is adopted for the column generation approach. Numerical results on four transportation networks demonstrate the superior efficiency and robustness of the GP algorithm with the BB step size scheme over the self-adaptive GP method.

Escort Function Definitions for Feasible Region Flexibility in Applications of Replicatorlike Dynamics With Constraints

This articleproposes an extension to a family of evolutionary game dynamics called escort replicator dynamics (ERD), defined by (Harper, 2011) based on information-geometric concepts. Specifically, the objective of this article is to propose three types of convenient so-called escort functionsand evaluate the stability of ERD under the use of these functions. Since it is proven that ERD is stable using the proposed definitions, ERD is able to confine trajectories of the state vector in regions defined by upper and lower boundaries which can be, but are not required to be, nonnegative. From an engineering application perspective, these proposed escort functions are convenient since they allow to extend replicatorlike dynamics applications to engineering problems with feasible regions more general than the standard simplex or the nonnegative orthant. From a population dynamics perspective the proposed escort functions are useful to represent upper and lower limits on the share of a population that is allowed to play a given pure strategy of the underlying symmetric game.

Recursive rank one perturbations for pole placement and cone reachability

The role of rank one perturbations in transforming the eigenstructure of a matrix has long been considered in the context of applications, especially in linear control systems. Two cases are examined in this paper: First, we propose a practical method to place the system eigenvalues (poles) in desired locations via feedback control that is computed in terms of recursive rank one perturbations. Second, a choice of feedback control is proposed in order to achieve that a trajectory eventually enters the nonnegative orthant and remains therein for all time thereafter. The latter situation is achieved by imposing the strong Perron-Frobenius property and involves altering the eigenvalues, as well as left eigenvectors via rank one perturbations.

Sublinear Circuits for Polyhedral Sets

Sublinear circuits are generalizations of the affine circuits in matroid theory, and they arise as the convex-combinatorial core underlying constrained non-negativity certificates of exponential sums and of polynomials based on the arithmetic-geometric inequality. Here, we study the polyhedral combinatorics of sublinear circuits for polyhedral constraint sets. We give results on the relation between the sublinear circuits and their supports and provide necessary as well as sufficient criteria for sublinear circuits. Based on these characterizations, we provide some explicit results and enumerations for two prominent polyhedral cases, namely the non-negative orthant and the cube [− 1,1]n.

A nonzero-sum risk-sensitive stochastic differential game in the orthant

<p style='text-indent:20px;'>We study a nonzero-sum risk-sensitive stochastic differential game for controlled reflecting diffusion processes in the nonnegative orthant. We treat two cost evaluation criteria, namely, discounted cost and ergodic cost. Under certain assumptions, we establish the existence of Nash equilibria. Also, we completely characterize a Nash equilibrium for the ergodic cost criterion in the space of stationary Markov strategies.</p>