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In this paper, we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or lift of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices. We show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual. This generalizes a theorem of Yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix. Symmetric lifts of convex sets can also be characterized similarly. When the cones live in a family, our results lead to the definition of the rank of a convex set with respect to this family. We present results about this rank in the context of cones of positive semidefinite matrices. Our methods provide new tools for understanding cone lifts of convex sets.

In this research we introduce the concept of strong -convexity for set-valued functions defined on -convex subsets of real linear normed spaces, a variety of properties and examples of these functions are shown, an inclusion of Jensen type is also exhibited.

Every polyhedron can be decomposed into a Minkowski sum (or vector sum) of a bounded polyhedron and a polyhedral cone. This paper establishes similar statements for some classes of discrete sets in discrete convex analysis, such as integrally convex sets, L ♮ -convex sets, and M ♮ -convex sets.

In this work, analyzing the cosets of the subgroup in the group of L - convex sets is presented as a new and powerful tool in the topics of the convex analysis and abstract algebra. On L - convex sets, the properties of these cosets are proved mathematically. Most important theorem on a finite group of L - convex sets theory which is the Lagrange's Theorem has been proved. As well as, the mathematical proof of the quotient group of L - convex sets is presented.

Some properties and applications of generalized convex sets and generalized convex functions in multidimensional real, complex, and hypercomplex spaces are described.

L-convex sets are one of the most fundamental concepts in discrete convex analysis. Furthermore, the Minkowski sum of two L-convex sets, called L 2 -convex sets, is an intriguing object that is closely related to polymatroid intersection. This paper reveals the polyhedral description of an L 2 -convex set, together with the observation that the convex hull of an L 2 -convex set is a box-TDI polyhedron. Two different proofs are given for the polyhedral description. The first is a structural short proof, relying on the conjugacy theorem in discrete convex analysis, and the second is a direct algebraic proof, based on Fourier–Motzkin elimination. The obtained results admit natural graph representations. Implications of the obtained results in discrete convex analysis are also discussed.

In this paper, a new type of convexity is defined, namely, the left–right-(k,h-m)-p IVM (set-valued function) convexity. Utilizing the definition of this new convexity, we prove the Hadamard inequalities for noninteger Katugampola integrals. These inequalities generalize the noninteger Hadamard inequalities for a convex IVM, (p,h)-convex IVM, p-convex IVM, h-convex, s-convex in the second sense and many other related well-known classes of functions implicitly. An apt number of numerical examples are provided as supplements to the derived results.

Calculating the failure probability of a landslide is important in engineering related to geological process and geomorphological evolution. Strength parameters of a soil (i.e., cohesion and internal friction angle) are regarded as uncertain-but-bounded parameters. In this study, a new method is proposed for computing the landslide failure probability based on a convex set model and artificial neural network (ANN). In the new method, ANN is used to determine the limit state function of landslide stability, and the failure probability is determined using a simple iterative algorithm. The new method was applied to calculate the failure probability of the Gufenping landslide in Nanjiang, Sichuan, China. The results calculated using a Monte Carlo simulation (MCS) method confirmed that the new method accurately and quickly obtains the failure probability of a landslide. Additionally, compared with the two-dimensional calculation method, the one-dimensional analysis method overestimates the failure probability of the landslide. The results of single factor and global sensitivity analysis indicate that the average of internal friction angle is the main factor affecting the stability of the landslide. It is easy to calculate failure probability of landslides using the novel method than using the conventional methods.

In this research we lay the concept of log -convex functions defined on real intervals containing the origin, some algebraic properties are exhibit, in the same token discrete Jensen type inequalities and integral inequalities are set and shown.

The faces of the unit ball of a finite-dimensional Banach space are automatically closed. The situation is different in the infinite-dimensional case. In fact, under this last condition, the closure of a face may not be a face. In this paper, we discuss these issues in an expository style. In order to illustrate the described situation we consider an equivalent renorming of the Banach space ℓ1.