Integral Convex Research Articles

Asymptotics of Solution to Indirect Optimal Control Problem with Integral Convex Performance Index and Cheap Control

Asymptotics of Solution to Indirect Optimal Control Problem with Integral Convex Performance Index and Cheap Control

1-bit Massive MIMO Signal Detector Based on Convex Integral Quadratic Programming

1-bit Massive MIMO Signal Detector Based on Convex Integral Quadratic Programming

Distributed Integral Convex Optimization-Based Current Control for Power Loss Optimization in Direct Current Microgrids

Due to the advantages of fewer energy conversion stages and a simple structure, direct current (DC) microgrids are being increasingly studied and applied. To minimize distribution loss in DC microgrids, a systematic optimal control framework is proposed in this paper. By considering conduction loss, switching loss, reverse recovery loss, and ohmic loss, the general loss model of a DC microgrid is formulated as a multi-variable convex function. To solve the objective function, a top-layer distributed integral convex optimization algorithm (DICOA) is designed to optimize the current-sharing coefficients by exchanging the gradients of loss functions. Then, the injection currents of distributed energy resources (DERs) are allocated by the distributed adaptive control in the secondary control layer and local voltage–current control in the primary layer. Based on the DICOA, a three-layer control strategy is constructed to achieve loss minimization. By adopting a peer-to-peer data-exchange strategy, the robustness and scalability of the proposed systematic control are enhanced. Finally, the proposed distribution current dispatch control is implemented and verified by simulations and experimental results under different operating scenarios, including power limitation, communication failure, and plug-in-and-out of DERs.

Asymptotics of a Solution to an Optimal Control Problem with Integral Convex Performance Index, Cheap Control, and Initial Data Perturbations

Asymptotics of a Solution to an Optimal Control Problem with Integral Convex Performance Index, Cheap Control, and Initial Data Perturbations

Asymptotics of Solutions of Linear Singularly Perturbed Optimal Control Problems with a Convex Integral Performance Index and a Cheap Control

We consider an optimal control problem for a linear system with constant coefficients with an integral convex performance index containing a small parameter multiplying the integral term in the class of piecewise continuous controls with smooth geometric constraints. Such problems are called cheap control problems. It is shown that the limit problem will be a problem with a terminal performance index. It is established that if the terminal term of the performance index is a convex (strictly convex) and continuously differentiable function, then the performance functional in the limit problem has similar properties. It is proved that, in the general case, convergence with respect to the performance functional is valid, and under the condition of strict convexity of the terminal term of the performance index in the original problem, convergence to the minimum point of the terminal summand of the performance index in the limit problem is valid. The limit of the defining vector in the original problem is found as the small parameter tends to zero. In particular, it is shown that the first component of the defining vector in the original problem converges to the defining vector in the limit problem. The problems of controlling a point of low mass in a medium with and without resistance with a terminal part depending on both slow and fast variables are considered in detail, and complete asymptotic expansions of the defining vectors in these problems are constructed.

Asymptotics of Solutions of Linear Singularly Perturbed Optimal Control Problems with a Convex Integral Performance Index and a Cheap Control

Asymptotics of Solutions of Linear Singularly Perturbed Optimal Control Problems with a Convex Integral Performance Index and a Cheap Control

A further investigation of convex integrals based on monotone measures

The convex integral introduced by Mesiar et al. (Superdecomposition integrals, Fuzzy Sets Syst. 259 (2015) 3–11) is a special superdecomposition integral related to convex functional. It can be regarded as the counterpart of concave integral, but it is not fully dual to the concave integral. This paper investigates some properties of the convex integral in detail. We show which of these results are valid both for the concave and convex integrals, which are dual to each other, and which are valid only for one of these two integrals. We also provide a necessary and sufficient condition for which two concave integrals (resp. convex integral) based on different monotone measures coincide with each other.

Projective toric codes

Any integral convex polytope [Formula: see text] in [Formula: see text] provides an [Formula: see text]-dimensional toric variety [Formula: see text] and an ample divisor [Formula: see text] on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on [Formula: see text], obtained by evaluating global section of the line bundle corresponding to [Formula: see text] on every rational point of [Formula: see text]. This work presents an extension of toric codes analogous to the one of Reed–Muller codes into projective ones, by evaluating on the whole variety instead of considering only points with nonzero coordinates. The dimension of the code is given in terms of the number of integral points in the polytope [Formula: see text] and an algorithmic technique to get a lower bound on the minimum distance is described.

Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting

The $ \Gamma $-limit of a family of functionals $ u\mapsto \int_{\Omega }f\left( \frac{x}{\varepsilon },\frac{x}{\varepsilon ^{2}},D^{s}u\right) dx $ is obtained for $ s = 1,2 $ and when the integrand $ f = f\left( y,z,v\right) $ is a continous function, periodic in $ y $ and $ z $ and convex with respect to $ v $ with nonstandard growth. The reiterated two-scale limits of second order derivatives are characterized in this setting.

Directed discrete midpoint convexity

For continuous functions, midpoint convexity characterizes convex functions. By considering discrete versions of midpoint convexity, several types of discrete convexities of functions, including integral convexity, L^natural-convexity and global/local discrete midpoint convexity, have been studied. We propose a new type of discrete midpoint convexity that lies between L^natural-convexity and integral convexity and is independent of global/local discrete midpoint convexity. The new convexity, named DDM-convexity, has nice properties satisfied by L^natural-convexity and global/local discrete midpoint convexity. DDM-convex functions are stable under scaling, satisfy the so-called parallelogram inequality and a proximity theorem with the same small proximity bound as that for L^{natural }-convex functions. Several characterizations of DDM-convexity are given and algorithms for DDM-convex function minimization are developed. We also propose DDM-convexity in continuous variables and give proximity theorems on these functions.

CONVEXITY CONDITION OF THE GENERALIZED BERNATSKY INTEGRAL FOR ONE SUBCLASS OF STAR-LIKE FUNCTIONS

The article carried out a study on the convexity of the Bernatsky integral in the proposition that the function in question belongs to a subclass of star-shaped functions that satisfy certain conditions. For this, the condition of convexity of univalent functions was considered. The geometrical interpretation of the conditions is given, the radius of the convexity of the star-shaped functions is established. The intervals for the parameter are found for which the Bernatsky integral is a convex function in the whole unit circle, in cases where the parameter does not belong to the given interval, the Bernatsky integral will be a convex function in a circle of smaller radius. Three consequences are given in which various cases of convexity of the Bernatsky integral for analytic functions that belong to classes of functions with certain conditions are analyzed. For the considered classes of analytic functions, the radius of convexity of the Bernatsky integral is determined.

Explicit algorithms for J-fixed points of some non linear mappings in certain Banach spaces

Let E be a real normed linear space and E∗ its dual. In a recent work, Chidume et al. [Chidume, C. E. and Idu, K. O., Approximation of zeros of bounded maximal monotone mappings, solutions of hammerstein integral equations and convex minimizations problems, Fixed Point Theory and Applications, 97 (2016)] introduced the new concepts of J-fixed points and J-pseudocontractive mappings and they shown that a mapping A : E → 2 E∗ is monotone if and only if the map T := (J −A) : E → 2 E∗ is J-pseudocontractive, where J is the normalized duality mapping of E. It is our purpose in this work to introduce an algorithm for approximating J-fixed points of J-pseudocontractive mappings. Our results are applied to approximate zeros of monotone mappings in certain Banach spaces. The results obtained here, extend and unify some recent results in this direction for the class of maximal monotone mappings in uniformly smooth and strictly convex real Banach spaces. Our proof is of independent interest.

Admissibilization for Implicit Jump Systems With Mixed Retarded Delays Based on Reciprocally Convex Integral Inequality and Barbalat’s Lemma

This article considers admissibility analysis and stabilization for implicit Markovian jump systems (IMJSs) with retarded discrete-distributed delays. Admissibility analysis is investigated for the unforced delay IMJSs by virtue of reciprocally convex integral inequality technique and Barbalat’s lemma. State feedback controller is designed via the matrix transformation technique to realize the stabilization of the delayed closed-loop IMJSs. By selecting comprehensive L-K functional with modes and delays information, admissibilization conditions are presented in terms of LMIs. Two illustrative examples including an inverted pendulum controlled by a direct current motor (DCMCIP) system are utilized to certify the effectiveness and practicality of the admissibilization technique.

Integral Convexity and Parabolic Systems

In this work we give optimal, i.e., necessary and sufficient, conditions for integrals of the calculus of variations to guarantee the existence of solutions---both weak and variational solutions---...

The Trade-Off Between Privacy and Fidelity via Ehrhart Theory

As an increasing amount of data is gathered nowadays and stored in databases (DBs), the question arises of how to protect the privacy of individual records in a DB even while providing accurate answers to queries on the DB. Differential Privacy (DP) has gained acceptance as a framework to quantify vulnerability of algorithms to privacy breaches. We consider the problem of how to sanitize an entire DB via a DP mechanism, on which unlimited further querying is performed. While protecting privacy, it is important that the sanitized DB still provide accurate responses to queries. The central contribution of this work is to characterize the amount of information preserved in an optimal DP DB sanitizing mechanism (DSM). We precisely characterize the utility-privacy trade-off of mechanisms that sanitize DBs in the asymptotic regime of large DBs. We study this in an information-theoretic framework by modeling a generic distribution on the data, and a measure of fidelity between the histograms of the original and sanitized DBs. We consider the popular $\mathbb{L}_{1}-$distortion metric that leads to the formulation as a linear program (LP). This optimization problem is prohibitive in complexity with the number of constraints growing exponentially in the parameters of the problem. Leveraging tools from discrete geometry, analytic combinatorics, and duality theorems of optimization, we fully characterize the optimal solution in terms of a power series whose coefficients are the number of integer points on a multidimensional convex polytope studied by Ehrhart in 1967. Employing Ehrhart theory, we determine a simple closed form computable expression for the asymptotic growth of the optimal privacy-fidelity trade-off to infinite precision. At the heart of the findings is a deep connection between the minimum expected distortion and the Ehrhart series of an integral convex polytope.

Improved results on synchronisation of Markovian jump complex dynamical networks via sampled-data controller and convex combination

This paper investigates the Markovian jump complex dynamical networks via a sampled-data controller and convex combination. Firstly, constructing an improved discontinuous Lyapunov-Krasovskii function (LKF), which is fully considered the characteristics of sampled-data to reduce the conservativeness. Secondly, designing an appropriate sample-data controller and combining some integral inequalities, convex function and free weighting matrices to utilise the upper bound on variable sampling interval and the sawtooth structure information of varying input delay. Then one can obtain the new criterion of sampled-data synchronisation. Finally, giving two examples to prove the effectiveness and superiority of the proposed methods. Obviously, compared with other existing literatures, this paper has less conservative.

Relationship between two types of superdecomposition integrals on finite spaces

This paper investigates the relationship between two types of superdecomposition integrals, namely, the convex integral and the pan-integral from above, on finite spaces. To this end, we introduce two new concepts related to monotone measures — superadditivity with respect to singletons and minimal strictly subadditive set — and discuss some of their properties. In the case that the monotone measure μ is superadditive with respect to singletons, we show that these two types of integrals are equivalent. In other cases, by means of the characteristics of minimal strictly subadditive sets we provide a set of necessary and sufficient conditions for which these two types of integrals coincide with each other.

Qualification Conditions-Free Characterizations of the $$\varepsilon $$-Subdifferential of Convex Integral Functions

We provide formulae for the \(\varepsilon \)-subdifferential of the integral function \(I_f(x):=\int _T f(t,x) d\mu (t)\), where the integrand \(f:T\times X \rightarrow \overline{\mathbb {R}}\) is measurable in (t, x) and convex in x. The state variable lies in a locally convex space, possibly non-separable, while T is given a structure of a nonnegative complete \(\sigma \)-finite measure space \((T,\mathcal {A},\mu )\). The resulting characterizations are given in terms of the \(\varepsilon \)-subdifferential of the data functions involved in the integrand, f, without requiring any qualification conditions. We also derive new formulas when some usual continuity-type conditions are in force. These results are new even for the finite sum of convex functions and for the finite-dimensional setting.

Discrete Midpoint Convexity

For a function defined on the integer lattice, we consider discrete versions of midpoint convexity, which offer a unifying framework for discrete convexity of functions, including integral convexity, L♮-convexity, and submodularity. By considering discrete midpoint convexity for all pairs at ℓ∞-distance equal to 2 or not smaller than 2, we identify new classes of discrete convex functions, called locally and globally discrete midpoint convex functions. These functions enjoy nice structural properties. They are stable under scaling and addition and satisfy a family of inequalities named parallelogram inequalities. Furthermore, they admit a proximity theorem with the same small proximity bound as that for L♮-convex functions. These structural properties allow us to develop an algorithm for the minimization of locally and globally discrete midpoint convex functions based on the proximity-scaling approach and on a novel 2-neighborhood steepest descent algorithm.

Ehrhart Polynomials with Negative Coefficients

It is shown that, for each $$d \ge 4$$ , there exists an integral convex polytope $${\mathcal {P}}$$ of dimension d such that each of the coefficients of $$n, n^{2}, \ldots , n^{d-2}$$ of its Ehrhart polynomial $$i({\mathcal {P}},n)$$ is negative. Moreover, it is also shown that for each $$d \ge 3$$ and $$1 \le k \le d-2$$ , there exists an integral convex polytope $${\mathcal {P}}$$ of dimension d such that the coefficient of $$n^k$$ of the Ehrhart polynomial $$i({\mathcal {P}},n)$$ of $${\mathcal {P}}$$ is negative and all its remaining coefficients are positive. Finally, we consider all the possible sign patterns of the coefficients of the Ehrhart polynomials of low dimensional integral convex polytopes.