Convex Polynomial Research Articles

Derivation of New Degrees for Best COCUNP Weighted Approximation: II

Approximation Theory is a branch of analysis and applied mathematics requiring that the approximation process preserves certain <img src=image/13417154_01.gif>-shaped properties defined at a finite interval <img src=image/13417154_02.gif>, such as convexity in all or parts of the interval. The (Co)convex and Unconstrained Polynomial (COCUNP) approximation is one of the key estimations of the approximation theory that Kopotun has recently raised for ten years. Numerous studies have been conducted on modern methods of weighted approximation to construct the best degree of approximation. In developing COCUNP a novel technique, the Lebesgue Stieltjes integral-i technique is used to resolve certain disadvantages, such as Riemann's integrable functions, which do not have the degree of the best approximation in norm space. In order to achieve the main goal, Derivation of New Degree (DOND) of the best COCUNP approximation was constructions. The theoretical results revealed that, in general, the new degrees of best approximation were able to smaller errors compared to the existing literature in the same estimating. In conclusion, this study has successfully developed DOND for the best (Co)convex Polynomial (COCP) weighted approximation.

On solving a class of fractional semi-infinite polynomial programming problems

In this paper, we study a class of fractional semi-infinite polynomial programming (FSIPP) problems, in which the objective is a fraction of a convex polynomial and a concave polynomial, and the constraints consist of infinitely many convex polynomial inequalities. To solve such a problem, we first reformulate it to a pair of primal and dual conic optimization problems, which reduce to semidefinite programming (SDP) problems if we can bring sum-of-squares structures into the conic constraints. To this end, we provide a characteristic cone constraint qualification for convex semi-infinite programming problems to guarantee strong duality and also the attainment of the solution in the dual problem, which is of its own interest. In this framework, we first present a hierarchy of SDP relaxations with asymptotic convergence for the FSIPP problem whose index set is defined by finitely many polynomial inequalities. Next, we study four cases of the FSIPP problems which can be reduced to either a single SDP problem or a finite sequence of SDP problems, where at least one minimizer can be extracted. Then, we apply this approach to the four corresponding multi-objective cases to find efficient solutions.

A sufficient condition for asymptotically well behaved property of convex polynomials

In this paper, we employ an asymptotic assumption to show that the maximum of finitely many convex polynomials is an asymptotically well behaved function. We also give two examples to illustrate our main result. As an application, we reproduce some known results appeared recently in the literature.

Exact dual semi-definite programs for affinely adjustable robust SOS-convex polynomial optimization problems

This paper presents exact dual semi-definite programs (SDPs) for robust SOS-convex polynomial optimization problems with affinely adjustable variables in the sense that the optimal values of the robust problem and its associated dual SDP are equal with the solution attainment of the dual problem. This class of robust convex optimization problems includes the corresponding quadratically constrained convex quadratic optimization problems and separable convex polynomial optimization problems, and it employs a general bounded spectrahedron uncertainty set that covers the most commonly used uncertainty sets of numerically solvable robust optimization models, such as boxes, balls and ellipsoids. As special cases, it also demonstrates that explicit exact dual SDP and second-order cone programming (SOCP) in terms of original data hold for the robust two-stage convex quadratic programs with quadratic constraints and the robust two-stage separable convex quadratic programs under an ellipsoidal uncertainty set, respectively. Finally, the paper illustrates the results via numerical implementations of the developed SDP duality scheme on adjustable robust lot-sizing problems with nonlinear costs under demand uncertainty.

Thermal conductivity of undisturbed soil – Measurements and predictions

Soil thermal conductivity, λ, is decisive for a range of soil ecosystem services related to soil temperature. Soil functions affected by temperature comprise microbial as well as abiotic processes of importance to crop growth, leaching of nutrients, carbon sequestration, and emission of greenhouse gases. Models for crop production and climate change include λ for simulation of heat transfer in the natural environment. Nevertheless, most prediction equations for λ are based on measurements deriving from physically disturbed soil. We identified nine Danish soils with a gradient in clay from ~ 0.03 to 0.35 kg kg−1 and measured λ in situ at six soil depths from 5 to 85 cm with the heat pulse method. Soil cores were collected and used for measurements of λ at a range of moisture conditions in the laboratory. The degree of soil water saturation, S, ranged from 0.01 to 0.98, and measured λ was in the range 0.18–2.98 W m−1 K−1. A model for λ identified by multiple regression across all laboratory measurements explained 87% of the variation in data and pointed out a convex polynomial relation between S and λ. It included significant positive effects of bulk density (BD) and soil organic matter (SOM), while λ decreased with soil clay content for a given S. The stochastic model predicted λ for three independent data sets from the literature with little bias and root mean square errors in the range 0.15–0.23 W m−1 K−1. A polynomial regression of λ versus S was performed for combinations of site and soil depth. By this λ was estimated at completely dry conditions, λdry, and at the fully saturated state, λsat. The estimates of λdry as well as λsat had a wider range than reported in the literature. This study calls for re-evaluation of existing pedotransfer functions for λdry. Measurements of λ in soil of undisturbed structural conditions are encouraged.

Multi-objective convex polynomial optimization and semidefinite programming relaxations

This paper aims to find efficient solutions to a multi-objective optimization problem (MP) with convex polynomial data. To this end, a hybrid method, which allows us to transform problem (MP) into a scalar convex polynomial optimization problem \(({\mathrm{P}}_{z})\) and does not destroy the properties of convexity, is considered. First, we show an existence result for efficient solutions to problem (MP) under some mild assumption. Then, for problem \((P_{z})\), we establish two kinds of representations of non-negativity of convex polynomials over convex semi-algebraic sets, and propose two kinds of finite convergence results of the Lasserre-type hierarchy of semidefinite programming relaxations for problem \(({\mathrm{P}}_{z})\) under suitable assumptions. Finally, we show that finding efficient solutions to problem (MP) can be achieved successfully by solving hierarchies of semidefinite programming relaxations and checking a flat truncation condition.

On the solution existence to convex polynomial programs and its applications

In this paper, we present necessary/sufficient conditions for the convex polynomial programming (CPP) problems. Some new stability results for parametric CPP problems are characterized under a regular condition. We give a positive answer for the open question in Kim et al. (Optim Lett 6:363–373, 2012) for the solution existence of convex quadratic programming problems.

On Hermite-Hadamard type inequalities for $ n $-polynomial convex stochastic processes

In this note, our purpose is to introduce the concept of $ n $-polynomial convex stochastic processes and study some of their algebraic properties. We establish new refinements for integral version of Holder and power mean inequality. Also, we are concerned to extend several Hermite-Hadamard type inequalities for $ n $-polynomial convex stochastic processes by using Holder, Holder-Iscan, power mean and improved power mean integral inequalities. Moreover, we give comparison of obtained results.

$ (m, n) $-Harmonically polynomial convex functions and some Hadamard type inequalities on the co-ordinates

In this study, we have introduced a new concept called $ (m, n) $-harmonically polynomial convex functions on the co-ordinates. Then, we have demonstrated some properties of this definition. Based on the definition and some elementary analysis process, we have proved a new Hadamard type integral inequality on the coordinates for $ (m, n) $-harmonically polynomial convex functions. Finally, we have established Hadamard type inequality for differentiable $ (m, n) $-Harmonically polynomial convex functions. We have also given some special cases for bounded functions.

Fractional inequalities of the Hermite–Hadamard type for $ m $-polynomial convex and harmonically convex functions

In this paper, it is our purpose to establish some new fractional inequalities of the Hermite–Hadamard type for the $ m $-polynomial convex and harmonically convex functions. Our results involve the Caputo–Fabrizio and $ \zeta $-Riemann–Liouville fractional integral operators. They generalize, complement and extend existing results in the literature. By taking $ m\geq 2 $, we deduce loads of new and interesting inequalities. We expect that the thought laid out in this work will provoke advance examinations in this course.

Delay‐dependent robust control of stochastic systems with convex polynomial uncertainty

SummaryThis article addresses a robust control problem of time‐varying stochastic systems with time‐delays. Through Linear Parameter Varying (LPV) modeling approach, the time‐varying parameters can be described via the convex combination. Therefore, the LPV stochastic system is interpreted by a weighting function and multiplicative noised linear systems. Furthermore, stabilization problem for the systems is investigated via Gain‐Scheduled (GS) technique to increase robustness. For the problem, some sufficient conditions are derived by Integral Lyapunov‐Krasovskii Function (ILKF) such that the conservatism caused by derivative of weighting function is ignored. Besides, an Iterative Linear Matrix Inequality algorithm is used to determine the auxiliary variable such that the problem can be solved via the convex calculation. Finally, some simulations are provided to demonstrate effectiveness and applicability of this article.

Interpolatory pointwise estimates for convex polynomial approximation

This paper deals with approximation of smooth convex functions f on an interval by convex algebraic polynomials which interpolate f and its derivatives at the endpoints of this interval. We call such estimates “interpolatory”. One important corollary of our main theorem is the following result on approximation of $$f \in \Delta ^{(2)}$$ , the set of convex functions, from $$W^r$$ , the space of functions on $$[-1, 1]$$ for which $$f^{(r-1)}$$ is absolutely continuous and $$\Vert f^{(r)}\Vert_\infty := \mathrm{ess~sup}_{x\in [-1,1]}|f^{(r)}(x)| < \infty $$ : For any $$f \in W^{r} \cap \Delta ^{(2)}, r \in \mathbb {N}$$ , there exists a number $$ \mathcal {N} =\mathcal {N}(f, r)$$ , such that for every $$n \ge \mathcal {N}$$ , there is an algebraic polynomial of degree $$ \le n$$ which is in $$\Delta ^{(2)}$$ and such that $$\begin{aligned}{\left\| \frac{f-P_n}{\varphi ^{r}}\right\| }_{\infty } \le \frac{c(r)}{n^r}\Vert f^{(r)}\Vert _\infty ,\end{aligned}$$ where $$\varphi (x) := \sqrt{1 - x^2}$$ . For $$r = 1$$ and $$r = 2$$ , the above result holds with $$\mathcal {N} = 1$$ and is well known. For $$ r\ge 3$$ , it is not true, in general, with $$\mathcal {N}$$ independent of f.

Second-order cone programming relaxations for a class of multiobjective convex polynomial problems

This paper is concerned with a multiobjective convex polynomial problem, where the objective and constraint functions are first-order scaled diagonally dominant sums-of-squares convex polynomials. We first establish necessary and sufficient optimality criteria in terms of second-order cone (SOC) conditions for (weak) efficiencies of the underlying multiobjective optimization problem. We then show that the obtained result provides us a way to find (weak) efficient solutions of the multiobjective program by solving a scalar second-order cone programming relaxation problem of a given weighted-sum optimization problem. In addition, we propose a dual multiobjective problem by means of SOC conditions to the multiobjective optimization problem and examine weak, strong and converse duality relations.

On semi-infinite systems of convex polynomial inequalities and polynomial optimization problems

We consider the semi-infinite system of polynomial inequalities of the form $$\begin{aligned} {{\mathbf {K}}}:=\{x\in {{\mathbb {R}}}^m\mid p(x,y)\ge 0,\quad \forall y\in S\subseteq {{\mathbb {R}}}^n\}, \end{aligned}$$where p(x, y) is a real polynomial in the variables x and the parameters y, the index set S is a basic semialgebraic set in $${{\mathbb {R}}}^n$$, $$-p(x,y)$$ is convex in x for every $$y\in S$$. We propose a procedure to construct approximate semidefinite representations of $${{\mathbf {K}}}$$. There are two indices to index these approximate semidefinite representations. As two indices increase, these semidefinite representation sets expand and contract, respectively, and can approximate $${{\mathbf {K}}}$$ as closely as possible under some assumptions. In some special cases, we can fix one of the two indices or both. Then, we consider the optimization problem of minimizing a convex polynomial over $${{\mathbf {K}}}$$. We present an SDP relaxation method for this optimization problem by similar strategies used in constructing approximate semidefinite representations of $${{\mathbf {K}}}$$. Under certain assumptions, some approximate minimizers of the optimization problem can also be obtained from the SDP relaxations. In some special cases, we show that the SDP relaxation for the optimization problem is exact and all minimizers can be extracted.

Деякі негативні результати для інтерполяційного монотонного наближення функцій, що мають дробову похідну

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function $f є W^r [0,1]$ by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function $f є C^r [0,1] \cap \Delta^0$ where $\Delta^0$ is the set of positive functions on [0,1]. Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider $r є , r &gt; 2$. In [8] is consider $r є , r &gt; 2$. It was proved that for monotone approximation estimates of the form (1) are fails for $r є , r &gt; 2$. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6]). In [5] is consider $r є , r &gt; 2$. In [6] is consider $r є , r &gt; 2$. It was proved that for convex approximation estimates of the form (1) are fails for $r є , r &gt; 2$. In this paper the question of approximation of function $f є W^r \cap \Delta^1, r є (3,4)$ by algebraic polynomial $p_n є \Pi_n \cap \Delta^1$ is consider. The main result of the work generalize the result of work [8] for $r є (3,4)$.

Convex Stone-Weierstrass theorems and invariant convex sets

A \textit {convex polynomial} is a convex combination of the monomials $\{1, x, x^2, \ldots \}$. This paper establishes that the convex polynomials on $\mathbb R$ are dense in $L^p(\mu )$ and weak$^*$ dense in $L^\infty (\mu )$ whenever $\mu $ is a compactly supported regular Borel measure on $\mathbb {R}$ and $\mu ([-1,\infty )) = 0$. It is also shown that the convex polynomials are norm dense in $C(K)$ precisely when $K \cap [-1, \infty ) = \emptyset $, where $K$ is a compact subset of the real line. Moreover, the closure of the convex polynomials on $[-1,b]$ is shown to be the functions that have a convex power series representation. A continuous linear operator $T$ on a locally convex space $X$ is \textit {convex-cyclic} if there is a vector $x \in X$ such that the convex hull of the orbit of $x$ is dense in $X$. The previous results are used to characterize which multiplication operators on various real Banach spaces are convex-cyclic. Also, it is shown for certain multiplication operators that every nonempty closed invariant convex set is a closed invariant subspace.

"Узагальнення поточкових інтерполяційних оцінок опуклого наближення функцій, що мають дробову похідну "

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function of Sobolev`s space by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function of r times continuously differentiable and positive functions on [0, 1]. Estimates of the form (1) for positive approximation are known ([1, 2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3, 4, 8]. In [3, 4] is consider r is natural and r not equal one. In [8] is consider r is real and r more two. It was proved that for monotone approximation estimates of the form (1) are fails for r is real and r more two. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5, 6]). In [5] is consider r is natural and r not equal one. In [6] is consider r is real and r more two. It was proved that for convex approximation estimates of the form (1) are fails for r is real and r more two. In [9] the question of approximation of function of Sobolev`s space and convex by algebraic convex polynomial is consider. It is proved, that for this function, estimate (1) is not true, if r is more three and less four generally speaking. In this paper the question of approximation of function Sobolev`s space and convex by algebraic convex polynomial is consider. This paper is the generalization of results papers [9] and [11]. It is proved, that for function of Sobolev`s space and convex, estimate of the type (1) is not true, generally speaking. The main result is the ana­log of the theorem 2.3 in [11].

Implementation of a new constitutive model for abdominal muscles

Background and objectiveAbdominal hernia repair is one of the most often performed surgical procedures worldwide. Numerical simulations of the abdominal wall mechanics can be a valuable tool to devise actions aimed at preventing hernia formation. A first step towards this goal is the development of consistent constitutive models for the tissues that form the human abdominal wall. In this study we propose, for each of the tissues involved, a new formulation of the so–called transversely isotropic hyperelastic model (TIHM). MethodsWe propose a new TIHM for the human abdominal wall tissues and we present a systemic view of the methodology that we have implemented in the present study. First we consider the mathematical background of the TIHM. The novelty of our formulation is that both the isotropic and the fiber contributions to the strain energy function are characterized exclusively by polynomial convex functions of certain invariant quantities. Then, we provide a detailed description on how the constitutive model is implemented into an open source finite element (FE) software. In our approach we use the specific interface provided by the MFront software to incorporate our TIHM formulation into the Code Aster FE solver. For each of the tissues considered, the values of the TIHM constants are adjusted by means of a numerical simulation of previous experimental data from tensile tests. ResultsWe studied the following abdominal wall tissues: linea alba, rectus sheath, external oblique muscle, internal oblique muscle, transversus abdominis muscle and rectus abdominis muscle. Our formulation closely reproduces tensile test data for each tissue in the corresponding FE numerical simulation. ConclusionsThe new TIHM formulation is suitable for a future numerical investigation of the abdominal wall, which will in turn help us to assess the best zone to practice a colostomy. The methodology implemented in the present study can be easily extended in the future to develop and implement a TIHM for active muscles and/or a different type of constitutive model which might be suitable to characterize other tissues of biomedical interest.

On the complexity of detecting convexity over a box

It has recently been shown that the problem of testing global convexity of polynomials of degree four is strongly NP-hard, answering an open question of N.Z. Shor. This result is minimal in the degree of the polynomial when global convexity is of concern. In a number of applications however, one is interested in testing convexity only over a compact region, most commonly a box (i.e., a hyper-rectangle). In this paper, we show that this problem is also strongly NP-hard, in fact for polynomials of degree as low as three. This result is minimal in the degree of the polynomial and in some sense justifies why convexity detection in nonlinear optimization solvers is limited to quadratic functions or functions with special structure. As a byproduct, our proof shows that the problem of testing whether all matrices in an interval family are positive semidefinite is strongly NP-hard. This problem, which was previously shown to be (weakly) NP-hard by Nemirovski, is of independent interest in the theory of robust control.

Applying Gröbner Basis Method to Multiparametric Polynomial Nonlinear Programming

In this paper, we present first a new algorithm based on Grobner basis to analyze and/or solve a convex polynomial nonlinear programming problem that is a convex nonlinear programming in which the objective and constraints are algebraic polynomials. The main efficiency of our algorithm is that there is no need to compute feasible points to find the optimum value. Next, we generalize our results to analyze and/or solvemultiparametric convex polynomial nonlinear programming problems. The mainproperty of this method is that it preserves the parametric scheme of theproblem until the end of the algorithm. Even the output of our algorithm depends onparameters and so one can present the optimum value and optimizer points as afunction on parameters. To show the ability of this algorithm, we will state two applied examples: the problem of minimizing the expense of forage in a chicken farm with unpredictable price of forage and the number of chickens, and an optimal control problem in model predictive control. The presented algorithms in this paper are all implemented in the Maple software.