Linear Convex Research Articles

Convex linear combination of the controllability pairs for linear systems

Abstract The convex linear combination of the controllability pairs of linear continuous-time linear systems is defined and its properties are discussed. The main result is obtained using pure algebraic methods. In the illustrative examples different cases of linear convex combinations are analyzed.

PL DENSITY INVARIANT FOR TYPE II DEGENERATING K3 SURFACES, MODULI COMPACTIFICATION AND HYPER-KÄHLER METRIC

AbstractA protagonist here is a new-type invariant for type II degenerations of K3 surfaces, which is explicit piecewise linear convex function from the interval with at most $18$ nonlinear points. Forgetting its actual function behavior, it also classifies the type II degenerations into several combinatorial types, depending on the type of root lattices as appeared in classical examples.From differential geometric viewpoint, the function is obtained as the density function of the limit measure on the collapsing hyper-Kähler metrics to conjectural segments, as in [HSZ19]. On the way, we also reconstruct a moduli compactification of elliptic K3 surfaces by [AB19], [ABE20], [Brun15] in a more elementary manner, and analyze the cusps more explicitly.We also interpret the glued hyper-Kähler fibration of [HSVZ18] as a special case from our viewpoint, and discuss other cases, and possible relations with Landau–Ginzburg models in the mirror symmetry context.

Guest Editorial

Evolutionary Algorithms (EAs) are nature-inspired population-based search methods which work on Darwinian principles of natural selection. Due to their strong search capability and simplicity of implementation, EAs have been successfully applied to solve many complex optimization problems, which cannot be easily solved by traditional mathematical programming approaches, such as linear programming, quadratic programming, and convex optimization. Despite the great success enjoyed by EAs, it is worth noting that existing EA solvers usually conduct the search process from scratch, regardless of how similar the new problem encountered is to those already solved in the past. Therefore, conventional EAs do not learn from previous problems and the search capabilities of the EA solvers do not automatically grow with problem-solving experiences. However, in reality, since problems seldom exist in isolation, solving one problem may thus yield useful information for solving other related problems. In the literature, there is a growing interest in conducting research on evolutionary transfer optimization (ETO) in recent years: a paradigm that integrates EA solvers with knowledge learning and transfer across related domains to achieve better optimization efficiency and performance.

A study on neurosonography in neonates with hypoxic–ischemic encephalopathy and its correlation with neurodevelopmental outcome

Abstract Background: Hypoxic–ischemic encephalopathy (HIE) is a devastating condition causing severe neurologic deficits and deaths in children, occurring in about 1.5 out of 1000 live births. The pattern of brain injury depends on the severity and duration of hypoxia as well as the degree of brain maturation at the time of insult. The prognosis can depend largely on diagnosing the early screening of suspected cases, assessment of severity of the injury, and timely intervention. Aims and Objectives: This study is aimed at signifying the importance of neurosonography in neonates with HIE as a diagnostic tool and screening modality in the neonatal intensive care unit (NICU) and at establishing the role of neurosonography and Doppler sonography as an investigatory modality for predicting the neurological damage and influencing their neurodevelopmental outcome. Design: This is a prospective longitudinal-type study. Materials and Methods: A total of 50 neonates admitted in the NICU were included in the study from January 2017 to August 2018 with FUJIFILM SONOSITE (Bothell, Washington, USA) machine using a high-frequency linear probe (6–12 MHz) and convex probe (2–5 MHz). A total of 50 neonates admitted to the NICU were selected and enrolled in the study after fulfilling the selection criteria. The first scans were obtained within 72 h of birth, and subsequent follow-up scans were done between the 8th and 10th day and on the 30th day. The infants were then followed up after 6 to 12 months for a detailed neurological assessment. Statistical Analysis Used: All the statistical analyses were carried out using Fisher’s exact test. Results: Out of the eight neurosonographic (NSG) findings analyzed in our study that were found to be associated with hypoxemic brain changes, four of them showed a statistically significant correlation with high positive predictive value (PPV) with poor clinical outcome at 6 to 12 months of age. The PPV of neonates with ventriculomegaly, multicystic leukomalacia, abnormal Doppler indices, and intraventricular hemorrhage (IVH) was found to be 78%, 80%, 82%, and 87%, respectively. Conclusions: Neurosonography was found to be highly significant to predict the clinical outcome in neonates with HIE, and it should be used as the initial screening modality.

Mixed-Integer Convex Representability

Motivated by recent advances in solution methods for mixed-integer convex optimization (MICP), we study the fundamental and open question of which sets can be represented exactly as feasible regions of MICP problems. We establish several results in this direction, including the first complete characterization for the mixed-binary case and a simple necessary condition for the general case. We use the latter to derive the first nonrepresentability results for various nonconvex sets, such as the set of rank-1 matrices and the set of prime numbers. Finally, in correspondence with the seminal work on mixed-integer linear representability by Jeroslow and Lowe, we study the representability question under rationality assumptions. Under these rationality assumptions, we establish that representable sets obey strong regularity properties, such as periodicity, and we provide a complete characterization of representable subsets of the natural numbers and of representable compact sets. Interestingly, in the case of subsets of natural numbers, our results provide a clear separation between the mathematical modeling power of mixed-integer linear and mixed-integer convex optimization. In the case of compact sets, our results imply that using unbounded integer variables is necessary only for modeling unbounded sets.

Linear Convex Combination Estimators and Comparisons

In this paper, we introduce two linear convex combination estimators by using known estimators such as ordinary least squares, ridge and Liu estimators and examine the predictive performance of these estimators. Furthermore, a numerical example is examined to compare these estimators under the prediction mean squared error criterion.

Unconditionally energy stable second-order numerical scheme for the Allen\u2013Cahn equation with a high-order polynomial free energy

In this article, we consider a temporally second-order unconditionally energy stable computational method for the Allen–Cahn (AC) equation with a high-order polynomial free energy potential. By modifying the nonlinear parts in the governing equation, we have a linear convex splitting scheme of the energy for the high-order AC equation. In addition, by combining the linear convex splitting with a strong-stability-preserving implicit–explicit Runge–Kutta (RK) method, the proposed method is linear, temporally second-order accurate, and unconditionally energy stable. Computational tests are performed to demonstrate that the proposed method is accurate, efficient, and energy stable.

Convex optimization with an interpolation-based projection and its application to deep learning

Convex optimizers have known many applications as differentiable layers within deep neural architectures. One application of these convex layers is to project points into a convex set. However, both forward and backward passes of these convex layers are significantly more expensive to compute than those of a typical neural network. We investigate in this paper whether an inexact, but cheaper projection, can drive a descent algorithm to an optimum. Specifically, we propose an interpolation-based projection that is computationally cheap and easy to compute given a convex, domain defining, function. We then propose an optimization algorithm that follows the gradient of the composition of the objective and the projection and prove its convergence for linear objectives and arbitrary convex and Lipschitz domain defining inequality constraints. In addition to the theoretical contributions, we demonstrate empirically the practical interest of the interpolation projection when used in conjunction with neural networks in a reinforcement learning and a supervised learning setting.

A generic strategy to obtain semi‐analytical mesh sensitivities/velocities for tetrahedral mesh generators

AbstractThis article proposes a generic approach to obtain semi‐analytical mesh sensitivities and shows how to compute mesh updates for any tetrahedral mesher. The proposed strategy allows sensitivities to be computed for a given mesh or first improve an initial mesh and then compute sensitivities. The strategy is based on a modification of Persson and Strang's mesher, Distmesh, analogizing the mesh to a simple truss structure. The boundary of the mesh is maintained using Lagrangian multipoint constraints. An analytical relationship between the mesh and the boundary permits the nodes to move freely along the boundary. This links the mesh directly to the parameterized geometry. We demonstrate the approach's ability to produce accurate, smooth and continuous mesh sensitivities and update the mesh. We use the resulting sensitivities to predict mesh updates corresponding to geometry updates. Four examples are given that show the method can accurately describe linear and nonlinear convex and concave domains. An example is given that also demonstrates the ability to generate accurate mesh sensitivities for a mesh created using the third party mesher GMSH.

Max-plus convexity in Archimedean Riesz spaces

We study the topological properties of max-plus convex sets in an Archimedean Riesz space $E$ with respect to the topology and the max-plus structure associated to a given order unit $\boldsymbol u$; the definition of max-plus convex sets is algebraic and we do not assume that $E$ has an {\it a priori} given topological structure. To a given unit $\un$ one can associate two equivalent norms on $E$ one of which, denoted $\|\cdot\|_{\boldsymbol u}$, is classical, the other $\|\cdot\|_{\hun}$ is introduced here following a previous unpublished work of Stéphane Gaubert on the geodesic structure of finite dimensional max-plus; it is shown that the distance ${\sf D}_{\boldsymbol h\boldsymbol u}$ on $E$ associated to $\|\cdot\|_{\boldsymbol h\boldsymbol u}$ is a geodesic distance, called the Hilbert affine distance associated to $\boldsymbol u$, for which max-plus convex sets in $E$ are precisely the geodesically closed sets. Under suitable assumptions, we establish max-plus versions of some fixed points and continuous selection theorems that are well known for linear convex sets and we show that hyperspaces of compact max-plus convex sets are Absolute Retracts. We formulate a max-plus version of the Knaster-Kuratowski-Mazurkiewicz Lemma from which, following A. Granas and J. Dugundji, all of the consequences of the classical KKM Lemma can be derived in a max-plus version. P. de la Harpe showed that the interior of the standard simplex $\Delta_n$ equipped with the classical Hilbert metric-defined by the cross-ration of four appropriate points is isometric to a finite dimensional normed space. We give an explicit proof of that result: the norm space in question is $\mathbb R^n$ with the Hilbert affine norm $\|\cdot\|_{\boldsymbol h\boldsymbol u}$ with respect to $\boldsymbol u = (1, \ldots, 1)$.

Portfolio optimization under a minimax rule revisited

In this paper, we revisit the bi-criteria portfolio optimization model where the short selling is permitted, and a trade-off is sought between the expected return rate of a portfolio and the maximum of the uncertainty measured by a general deviation measure for all the investments comprising a portfolio. We solve this bi-criteria model by first converting it into a collection of weighted sum piecewise linear convex programs, and then analysing their optimality conditions. We not only provide explicit analytical formulas for all the efficient portfolios, but also explore as a whole the set of all the efficient portfolios and its structure such as dimensionality and distribution. We generalize the classical Two-fund Theorem by providing some collections of finitely many efficient portfolios to generate or estimate the set of all the efficient portfolios. We also notice that our efficient portfolios are almost the risk parity ones in the sense that the risks are allocated equally across the investments. Moreover, we illustrate the reliability of our model by carrying out Monte Carlo simulations to test the performance of some efficient portfolios versus inefficient ones.

Parameterized Algorithms for MILPs with Small Treedepth

Solving (mixed) integer (linear) programs, (M)I(L)Ps for short, is a fundamental optimisation task with a wide range of applications in artificial intelligence and computer science in general. While hard in general, recent years have brought about vast progress for solving structurally restricted, (non-mixed) ILPs: n-fold, tree-fold, 2-stage stochastic and multi-stage stochastic programs admit efficient algorithms, and all of these special cases are subsumed by the class of ILPs of small treedepth. In this paper, we extend this line of work to the mixed case, by showing an algorithm solving MILP in time f(a,d)poly(n), where a is the largest coefficient of the constraint matrix, d is its treedepth, and n is the number of variables. This is enabled by proving bounds on the denominators (fractionality) of the vertices of bounded-treedepth (non-integer) linear programs. We do so by carefully analysing the inverses of invertible sub-matrices of the constraint matrix. This allows us to afford scaling up the mixed program to the integer grid, and applying the known methods for integer programs. We then trace the limiting boundary of our "bounded fractionality" approach both in terms of going beyond MILP (by allowing non-linear objectives) as well as its usefulness for generalising other important known tractable classes of ILP. On the positive side, we show that our result can be generalised from MILP to MIP with piece-wise linear separable convex objectives with integer breakpoints. On the negative side, we show that going even slightly beyond such objectives or considering other natural related tractable classes of ILP leads to unbounded fractionality. Finally, we show that restricting the structure of only the integral variables in the constraint matrix does not yield tractable special cases.

RGB Colors and Ecological Optics

Object color space is highly structured due to optical constraints (radiant power non-negative, reflectance factors between zero and unity) and ecological context (daylight illuminant). In this setting trichromacy induces a natural geometry through a unique spectral tripartition. Different from null-context colorimetry, one gains two desirable relations: The colorimetric coordinates are coarse-grained spectral reflectance factors and there is a direct link to color experiences, since RGB–coordinates provide ostensive definitions. The framework allows one to deal with subtractive color mixture, source variation, effects of metamerism and relations between scenes and image data in a unified, structured manner. In ecological contexts, colors are effectively object properties. The formal framework is linear algebra and convex geometry. Applications in human biology, computer graphics, design, etc., are immediate.

Linear Signal Cancellation of Nonlinear Pulsing Schemes in a Verasonics Research Scanner.

Contrast-enhanced ultrasound (CEUS) is a real-time imaging technique that allows the visualization of organ and tumor microcirculation by utilizing the nonlinear response of microbubbles. Nonlinear pulsing schemes are used exclusively in CEUS imaging modes in modern scanners. One important aspect of nonlinear pulsing schemes is the near-complete elimination of the linear signals that originate from tissue. Up until now, no study has investigated the performance of Verasonics scanners in eliminating the linear signals during CEUS and, by extension, the optimal pulsing sequences for performing CEUS. The aim of this article was to investigate linear signal cancellation of the Verasonics scanner performing nonlinear pulsing schemes with two different probes (L7-4 linear array and C5-2 convex array). We have considered two pulsing schemes: pulse inversion (PI) and amplitude modulation (AM). We have also compared our results from the Verasonics scanner with a clinical scanner (Philips iU22). We found that the linear signal cancellation of the transmitted pulse by Verasonics scanner was ~40 dB in AM mode and ~30 dB in PI mode when operated at 0.06 MI. The linear signal cancellation performance of Verasonics scanner was comparable with Philips iU22 scanner in focused AM mode and on average 3 dB better than Philips iU22 scanner in focused PI mode.

Robust control for incremental quadratic constrained nonlinear time-delay systems subject to actuator saturation

This article presents a saturation controller design method for a type of nonlinear time-delay systems. Different from the current works, the nonlinear systems considered herein satisfy an incremental quadratic constraint, which is a more general nonlinearity. Firstly, for the constraint in the actuator, the convex set theory is employed to transform the saturated feedback law into a parameterized linear convex hull set. Next, the convex hull Lyapunov function is constructed to obtain sufficient conditions for exponential stability of the closed-loop system. In addition, by solving an optimization problem, the maximal estimation of domain of attraction is also determined. Finally, numerical examples are provided to verify the proposed method.

Competitive model of pumped storage power plants participating in electricity spot Market——in case of China

With the development of transmission and distribution price reform in China, pumped storage power station can not continue to be included in the effective assets of the power grid, and its cost can not be dredged through the transmission and distribution price, so it is urgent to find a way to protect its own income through the market. This paper innovatively proposes a “three-stage” competitive optimization model for pumped-storage power stations, using a quadratic programming algorithm with two consecutive iterations to convert the discrete programming problem into a linear convex programming problem, reducing the difficulty of calculation and improving the calculation accuracy. Finally, the reinforcement learning algorithm is used to obtain the real-time bidding strategy of the pumped storage power station, and continuous feedback is provided. The calculation example analysis shows that compared with the traditional model, the “three-stage” model can bring better benefits to the pumped storage power station, and when the actual value of demand fluctuates within −8%, the pumped storage power station has the ability to resist risks higher than the market average. And when the proportion of renewable energy increases from the current 8%–30%, the revenue of pumped storage power plants will drop by 20%.

Further Results on Stability Analysis for Uncertain Delayed Neural Networks with Reliable Memory Feedback Control

This study proposes further results for the stability analysis issue of uncertain delayed neural networks (UDNNs) via the reliable memory feedback control scheme. First, an improved quadratic function method is introduced for constructing a novel term V 1 x t , which can fully excavate some intrinsic relationships between the delay derivative information and time-delay information. Based on the time-delay-product function (TDPF) and linear convex combination method (LCCM), the information storage is further improved for obtaining new theoretical results. Second, by using resultful integral inequalities and correlation analysis approaches, several relaxed criteria are established with respect to the asymptotical stability of the considered UDNNs. Third, the desired reliable memory feedback controller (RMFC) is achieved, which can ensure the system stability of UDNNs. Lastly, two numerical experiments are given to illustrate the significance of the theoretical results.

Lagrangian approximations for stochastic reachability of a target tube

We consider the problem of stochastic reachability of a target tube for a discrete-time, stochastic dynamical system with bounded control authority. We propose grid-free algorithms to compute under- and over-approximations of the stochastic reach set, and a corresponding feedback-based controller associated with a given likelihood. Our approach employs set-theoretic (Lagrangian) techniques for nonlinear systems with affine stochastic disturbances. With available set computation tools, these algorithms can only be implemented on linear systems. We use the law of total probability to partition the potentially unbounded disturbance set, and compute minimal and maximal reach sets via a robust approach. We propose a heuristic to partition the disturbance set using an ellipsoid when the disturbance is Gaussian, and a polyhedron otherwise. For linear dynamics and convex and compact sets, we employ existing tools from support functions and robust linear programming to efficiently approximate the stochastic reach set. We synthesize an admissible controller using the disturbance minimal reach sets by solving a series of linear programs. We demonstrate our approach on systems with linear dynamics (time-invariant and time-varying) and arbitrary disturbances (as well as Gaussian), including trajectory following for a Dubin’s vehicle and space vehicle rendezvous and docking.

A Simple Benchmark Problem for the Numerical Methods of the Cahn–Hilliard Equation

We present a very simple benchmark problem for the numerical methods of the Cahn–Hilliard (CH) equation. For the benchmark problem, we consider a cosine function as the initial condition. The periodic sinusoidal profile satisfies both the homogeneous and periodic boundary conditions. The strength of the proposed problem is that it is simpler than the previous works. For the benchmark numerical solution of the CH equation, we use a fourth-order Runge–Kutta method (RK4) for the temporal integration and a centered finite difference scheme for the spatial differential operator. Using the proposed benchmark problem solution, we perform the convergence tests for an unconditionally gradient stable scheme via linear convex splitting proposed by Eyre and the Crank–Nicolson scheme. We obtain the expected convergence rates in time for the numerical schemes for the one-, two-, and three-dimensional CH equations.

Two methods to approximate the Koopman operator with a reservoir computer.

The Koopman operator provides a powerful framework for data-driven analysis of dynamical systems. In the last few years, a wealth of numerical methods providing finite-dimensional approximations of the operator have been proposed [e.g., extended dynamic mode decomposition (EDMD) and its variants]. While convergence results for EDMD require an infinite number of dictionary elements, recent studies have shown that only a few dictionary elements can yield an efficient approximation of the Koopman operator, provided that they are well-chosen through a proper training process. However, this training process typically relies on nonlinear optimization techniques. In this paper, we propose two novel methods based on a reservoir computer to train the dictionary. These methods rely solely on linear convex optimization. We illustrate the efficiency of the method with several numerical examples in the context of data reconstruction, prediction, and computation of the Koopman operator spectrum. These results pave the way for the use of the reservoir computer in the Koopman operator framework.