Convex Extension Research Articles

2D Seismic Structural Study of the Area Between Halfaya, Noor and Amara Oil Fields, Southeastern Iraq

The study area is located in the Missan government in southeastern Iraq, including oilfields (Halfaya, Noor, and Amara). The study area is about 966 Km2. Based on the 2D seismic reflection interpretation, well-logs data (sonic and estimated density logs), and synthetic seismogram of well Am-2, two horizons were identified and picked (Nahr Umr and Shuaiba) within the Lower Cretaceous age. For the picked reflectors, two-way time and depth maps were created to display the subsurface's structural makeup. The structural interpretation shows two structural axes in the study area. The first is the Halfaya-Amara axis, whose direction is WNW-ESE corresponding with the Amara structure. This axis deviates southeast due to the tectonic effect at the Halfaya structure. Another tectonic force affected the Halfaya structure in the northwest-southeast direction leading to the appearance of a branch of a convex extension towards the north from the middle of the Halfaya structure, completing this branch to a structural closure as nose towards the Noor structure in the northern part of the study area. The second axis of Halfaya-Noor oilfields trending northwest-southeast direction.

Positivity and convexity in incomplete cooperative games

Incomplete cooperative games generalize the classical model of cooperative games by omitting the values of some of the coalitions. This allows for incorporating uncertainty into the model and studying the underlying games and possible payoff distributions based only on the partial information. In this paper, we conduct a systematic investigation of incomplete games, focusing on two important classes: positive and convex games. Regarding positivity, we generalize previous results from a special class of minimal incomplete games to a general setting. We characterize the non-extendability to a positive game by the existence of a certificate and provide a description of the set of positive extensions using its extreme games. These results also enable the construction of explicit formulas for several classes of incomplete games with special structures. The second part deals with convexity. We begin with the case of non-negative, minimal incomplete games. We establish the connection between incomplete games and the problem of completing partial functions and, consequently, provide a characterization of extendability and a full description of the set of symmetric convex extensions. This set serves as an approximation of the set of convex extensions.

On the Existence and Properties of Convex Extensions of Boolean Functions

On the Existence and Properties of Convex Extensions of Boolean Functions

Strongly convex functions and extensions of related inequalities with applications to entropy

<abstract><p>We extended the Mercer inequlaity, Fejér-Hermite-Hadamard, and Jensen inequalities for strongly convex functions. Moreover, we obtained several results in information theory and mathematical analysis using obtained inequalities.</p></abstract>

Time optimal problems on Lie groups and applications to quantum control

<abstract><p>In this paper we introduce a natural compactification of a left (right) invariant affine control system on a semi-simple Lie group $ G $ in which the control functions belong to the Lie algebra of a compact Lie subgroup $ K $ of $ G $ and we investigate conditions under which the time optimal solutions of this compactified system are "approximately" time optimal for the original system. The basic ideas go back to the papers of R.W. Brockett and his collaborators in their studies of time optimal transfer in quantum control (<sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>, <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>). We showed that every affine system can be decomposed into two natural systems that we call horizontal and vertical. The horizontal system admits a convex extension whose reachable sets are compact and hence posess time-optimal solutions. We then obtained an explicit formula for the time-optimal solutions of this convexified system defined by the symmetric Riemannian pair $ (G, K) $ under the assumption that the Lie algebra generated by the control vector fields is equal to the Lie algebra of $ K $.</p> <p>In the second part of the paper we applied our results to the quantum systems known as Icing $ n $-chains (introduced in <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>). We showed that the two-spin chains conform to the theory in the first part of the paper but that the three-spin chains show new phenomena that take it outside of the above theory. In particular, we showed that the solutions for the symmetric three-spin chains studied by (<sup>[<xref ref-type="bibr" rid="b3">3</xref>]</sup>, <sup>[<xref ref-type="bibr" rid="b4">4</xref>]</sup>) are solvable in terms of elliptic functions with the solutions completely different from the ones encountered in the two-spin chains.</p></abstract>

On the Existence and Properties of Convex Extensions of Boolean Functions

В данной статье изучается задача существования выпуклого продолжения произвольной булевой функции $f(x_1,x_2,…,x_n)$ на множество $[0,1]^n$. Построено $f_C(x_1,x_2,…,x_n)$ - выпуклое продолжение произвольной булевой функции $f(x_1,x_2,…,x_n)$ на множество $[0,1]^n$. На основе одного конструированного выпуклого продолжения $f_C(x_1,x_2,…,x_n)$ доказано, что для любой булевой функции $f(x_1,x_2,…,x_n)$ существует бесконечно много функций, каждая из которых является ее выпуклым продолжением на $[0,1]^n$. Также конструктивно доказано, что для любой булевой функции $f(x_1,x_2,…,x_n)$ существует единственная функция $f_{DM}(x_1,x_2,…,x_n)$, являющаяся максимальной среди всех ее выпуклых продолжений на множество $[0,1]^n$. Библиография: 15 названий.

Enhancing phenomenological yield functions with data: Challenges and opportunities

The formulation of history-dependent material laws has been a significant research and industrial activity in solid mechanics for over a century. A large variety of models has been developed, tailored for the description of different families of materials. However, model selection for a specific problem is a delicate issue and there still remain open problems. In fact, the catalog of yield models is continuously being enriched by experts to meet new needs and tailor models to new experimental evidence. We propose here an alternative approach, that is a flexible model-data-driven plasticity formulation, and analyze its challenges and opportunities. A phenomenological yield model is locally improved by a data-driven correction term. The data-based correction component is described by a surrogate model built from machine-learning techniques. In this regard, the framework is versatile, as shown by the fact that similar performances are obtained with Support Vector Regression, Gaussian Process Regression, and Neural Networks. The convexity of model-data yield functions is guaranteed by employing convex extensions of the adopted machine-learning techniques. The proposed approach is substantiated by reproducing a highly anisotropic yield response with tension/compression asymmetries from a simple phenomenological model enhanced with a limited number of synthetic data points of yield onset. It is shown that a crucial role is played by both model and data, the former allowing to use a limited number of data and the latter significantly enhancing model performance.

Discrete Aleksandrov solutions of the Monge-Ampere equation

We make two relaxations of the Oliker-Prussner method for the Dirichlet problem for the Monge-Ampere equation. First we relax the convexity requirement and consider mesh functions which are only discrete convex. The second relaxation consists in using a finite stencil. The discrete nonlinear equations are solved with a damped Newton's method. We give two proofs of convergence of the resulting scheme for right-hand side a density, on domains which are convex and not necessarily strictly convex, under the assumption that the boundary data has a continuous convex extension. The first proof is based on the notion of Aleksandrov solution while the second uses viscosity solutions. For more information see https://ejde.math.txstate.edu/conf-proc/26/a2/abstr.html

Exposing some points of interest about non-exposed points of desirability

We study the representation of sets of desirable gambles by sets of probability mass functions. Sets of desirable gambles are a very general uncertainty model, that may be non-Archimedean, and therefore not representable by a set of probability mass functions. Recently, Cozman (2018) has shown that imposing the additional requirement of even convexity on sets of desirable gambles guarantees that they are representable by a set of probability mass functions. Already more that 20 years earlier, Seidenfeld et al. (1995) gave an axiomatisation of binary preferences—on horse lotteries, rather than on gambles—that also leads to a unique representation in terms of sets of probability mass functions. To reach this goal, they use two devices, which we will call ‘SSK–Archimedeanity’ and ‘SSK–extension’. In this paper, we will make the arguments of Seidenfeld et al. (1995) explicit in the language of gambles, and show how their ideas imply even convexity and allow for conservative reasoning with evenly convex sets of desirable gambles, by deriving an equivalence between the SSK–Archimedean natural extension, the SSK–extension, and the evenly convex natural extension.

Set characterizations and convex extensions for geometric convex-hull proofs

In the present work, we consider Zuckerberg’s method for geometric convex-hull proofs introduced in Zuckerberg (Oper Res Lett 44(5):625–629, 2016). It has only been scarcely adopted in the literature so far, despite the great flexibility in designing algorithmic proofs for the completeness of polyhedral descriptions that it offers. We suspect that this is partly due to the rather heavy algebraic framework its original statement entails. This is why we present a much more lightweight and accessible approach to Zuckerberg’s proof technique, building on ideas from Gupte et al. (Discrete Optim 36:100569, 2020). We introduce the concept of set characterizations to replace the set-theoretic expressions needed in the original version and to facilitate the construction of algorithmic proof schemes. Along with this, we develop several different strategies to conduct Zuckerberg-type convex-hull proofs. Very importantly, we also show that our concept allows for a significant extension of Zuckerberg’s proof technique. While the original method was only applicable to 0/1-polytopes, our extended framework allows to treat arbitrary polyhedra and even general convex sets. We demonstrate this increase in expressive power by characterizing the convex hull of Boolean and bilinear functions over polytopal domains. All results are illustrated with indicative examples to underline the practical usefulness and wide applicability of our framework.

Locally $C^{1,1}$ convex extensions of $1$-jets

Let E be an arbitrary subset of \mathbb{R}^n , and let f\colon E\to\mathbb{R} , G\colon E\to\mathbb{R}^n be given functions. We provide necessary and sufficient conditions for the existence of a convex function F\in C^{1,1}_{\textrm{loc}}(\mathbb{R}^n) such that F=f and \nabla F=G on E . We give a useful explicit formula for such an extension F , and a variant of our main result for the class C^{1, \omega}_{\textrm{loc}} , where \omega is a modulus of continuity. We also present two applications of these results, concerning how to find C^{1,1}_{\textrm{loc}} convex hypersurfaces with prescribed tangent hyperplanes on a given subset of \mathbb{R}^n , and some explicit formulas for (not necessarily convex) C^{1,1}_{\textrm{loc}} extensions of 1 -jets.

Nonconvex noncash risk measures

In 2018, Cong and Zhao introduced noncash risk measures with 1-norm for weak cone-type acceptable sets. In the present paper, we take a further look at nonconvex, noncash risk measures with p-norm (1 ≤ p ≤ ∞) for nonweak cone-type acceptable sets. A convex extension of the nonconvex, noncash risk measures is given and a representation theorem of a convex extension of the nonconvex, noncash risk measures is achieved.

Theory and Methods of Euclidian Combinatorial Optimization: Current Status and Prospects

Euclidean combinatorial optimization problems are considered as discrete optimization problems on a set of combinatorial configurations mapped into an arithmetic Euclidean space. Modern methods of Euclidean combinatorial optimization are reviewed. The properties of the corresponding images of combinatorial sets are described. A theory of continuous functional representations and convex extensions is proposed for solving this class of problems. Areas of practical application and promising research areas are indicated.

Extensions of convex functions with prescribed subdifferentials

Let $E$ be an arbitrary subset of a Banach space $X$, $f: E \rightarrow \mathbb {R}$ a function, and $G:E \rightrightarrows X^*$ a set-valued mapping. We give necessary and sufficient conditions on $f, G$ for the existence of a continuous convex extension

Smooth convex extensions of convex functions

Let C be a compact convex subset of $$\mathbb {R}^n$$ , $$f:C\rightarrow \mathbb {R}$$ be a convex function, and $$m\in \{1, 2, \ldots , \infty \}$$ . Assume that, along with f, we are given a family of polynomials satisfying Whitney’s extension condition for $$C^m$$ , and thus that there exists $$F\in C^{m}(\mathbb {R}^n)$$ such that $$F=f$$ on C. It is natural to ask for further (necessary and sufficient) conditions on this family of polynomials which ensure that F can be taken to be convex as well. We give a satisfactory solution to this problem in the case $$m=\infty $$ , and make some remarks about the case of finite $$m\ge 2$$ . For a solution to a similar problem in the case $$m=1$$ (even for C not necessarily convex), see [1–3].

Passivity Based Control method for the diffusion process

Abstract A passivity based control method is presented for the diffusion process by the use of the concept of available storage and irreversible thermodynamics. A convex extension using the internal energy as generating function will be employed to define a Lyapunov functional for the distributed irreversible port Hamiltonian system (the diffusion process). The latter functional, will be used to suggest a Lyapunov stability condition by applying the so called La Salle’s Invariance principle for infinite dimensional systems on the heat and mass diffusion process.

Generalized Kohn-Sham iteration on Banach spaces.

A detailed account of the Kohn-Sham (KS) algorithm from quantum chemistry, formulated rigorously in the very general setting of convex analysis on Banach spaces, is given here. Starting from a Levy-Lieb-type functional, its convex and lower semi-continuous extension is regularized to obtain differentiability. This extra layer allows us to rigorously introduce, in contrast to the common unregularized approach, a well-defined KS iteration scheme. Convergence in a weak sense is then proven. This generalized formulation is applicable to a wide range of different density-functional theories and possibly even to models outside of quantum mechanics.

Non-Cash Risk Measure on Nonconvex Sets

Monetary risk measures defined on a convex set are interpreted as the smallest amount of external cash that must be added to a portfolio to make the portfolio being acceptable. In the present paper, the authors introduce a new concept: non-cash risk measure, which does as a nonconvex risk measure work in a nonconvex set. In addition, the authors arrive at a convex extension of the non-cash risk measure, and offer the relationship between the non-cash risk measure and its extension.

A note on stable Helmholtz decompositions in 3D

ABSTRACTThe stability of Helmholtz decompositions in 3D is known to hold for convex Lipschitz-continuous polyhedral regions and for arbitrary (not necessarily convex) domains of class . In this note we extend this result to non-convex Lipschitz-continuous polyhedral regions and to the case of homogeneous Neumann boundary conditions on a part of the boundary that is contained in the boundary of a convex extension of the original region. In addition, the relation with very similar results already available in the literature is also discussed. Finally, some implications of our results on the associated discrete Helmholtz decomposition and its application to the derivation of a posteriori error estimates, are briefly described.

Minimal convex extensions and finite difference discretisation of the quadratic Monge–Kantorovich problem

We present an adaptation of the Monge–Ampère (MA) lattice basis reduction scheme to the MA equation with second boundary value condition, provided the target is a convex set. This yields a fast adaptive method to numerically solve the optimal transport (OT) problem between two absolutely continuous measures, the second of which has convex support. The proposed numerical method actually captures a specific Brenier solution which is minimal in some sense. We prove the convergence of the method as the grid step size vanishes and show with numerical experiments that it is able to reproduce subtle properties of the OT problem.