Convex Analysis Research Papers - Academia.edu (original) (raw)

this paper, we look at the relationship between measurability so defined and the Borel measurability of Gamma, viewed as a single-valued function into the nonempty closed subsets of X, equipped with either the Hausdorff metric topology or... more

this paper, we look at the relationship between measurability so defined and the Borel measurability of Gamma, viewed as a single-valued function into the nonempty closed subsets of X, equipped with either the Hausdorff metric topology or with the AttouchWets topology. Our analysis rests on cardinality arguments in conjunction with the representation of these hyperspaces as weak topologies. Applications are given to convex-valued multifunctions.

This paper discusses an approach to solve the joint replenishment problem in a production environment with variable production cost. These variable production costs occur due to economies of scale in production. Under this environment,... more

This paper discusses an approach to solve the joint replenishment problem in a production environment with variable production cost. These variable production costs occur due to economies of scale in production. Under this environment, the model leads to a global optimization problem, which is investigated by using some standard results from convex analysis. Consequently, an effective and exact solution procedure is proposed. The proposed procedure is guaranteed to return a solution with a predetermined quality in terms of the objective function value. A computational study is provided to illustrate the performance of the proposed solution procedure with respect to the running time.

We introduce the notion of reciprocally strongly convex functions and we present some examples and properties of them. We also prove that two real functions f and g, defined on a real interval [a, b], satisfy f (xy/( tx + (1 − t)y)) ≤ tg... more

We introduce the notion of reciprocally strongly convex functions and we present some examples and properties of them. We also prove that two real functions f and g, defined on a real interval [a, b], satisfy f (xy/( tx + (1 − t)y)) ≤ tg (y) + (1 − t)g (x) − ct(1 − t) (1/ x − 1/ y)^ 2 , for all x, y ∈ [a, b] and t ∈ [0, 1] iff there exists a reciprocally strongly convex function h : [a, b] → R such that f (x) ≤ h (x) ≤ g (x) for all x ∈ [a, b]. Finally, we obtain an approximate convexity result for reciprocally strongly convex functions; namely we prove a stability result of Hyers-Ulam type for this class of functions.

Currently the modeling of check valves and flow control valves in water distribution systems is based on heuristics intermixed with solving the set of non-linear equations governing flow in the network. At the beginning of a simulation,... more

Currently the modeling of check valves and flow control valves in water distribution systems is based on heuristics intermixed with solving the set of non-linear equations governing flow in the network. At the beginning of a simulation, the operating status of these valves is not known and must be assumed. The system is then solved. The status of the check valves and flow control valves are then changed to try to determine their correct operating status, at times leading to incorrect solutions even for simple systems. This paper proposes an entirely different approach. Content and Co-Content theory is used to define conditions that lead to the guarantee of the existence and uniqueness of the solution. The work here focuses solely on flow control devices with a defined head discharge versus head loss relationship. A new modeling approach for water distribution systems based on subdifferential analysis that deals with the non-differentiable flow versus head relationships is proposed in this paper. The water distribution equations are solved as a constrained non-linear programming problem based on the Content model where the Lagrangian multipliers have important physical meanings. This new method gives correct solutions by dealing appropriately with inequality and equality constraints imposed by the presence of the flow regulating devices (check valves, flow control valves and temporarily closed isolating valves). An example network is used to illustrate the concepts.

Complex reaction networks (CRN) prove to be excellent models for complex chemical systems like chemical reactors or cellular compartments. Dynamics of the modeled sys- tem can be studied qualitatively using graph theoretical methods and... more

Complex reaction networks (CRN) prove to be excellent models for complex chemical systems like chemical reactors or cellular compartments. Dynamics of the modeled sys- tem can be studied qualitatively using graph theoretical methods and convex analysis. In this work, methodology for qualitative analysis of CRN is developed and imple- mented in MATLAB/Octave. Emphasis is put on comfort of the user and developer, i.e. on straightforward usage and lucidity of the code. Program for decomposition of a network into extreme pathways and for determination of their stability is implemented based on literature. Algorithm for automatic classification of potential oscillators is invented and an efficient genetic algorithm for finding Hopf bifurcation is proposed. The developed software is used to analyze 5 representative relevant CRN models.

We present the notion of reciprocally (s,m)-convex functions and present some examples and properties of them. We derive some inequalities for this new class of functions, specifically these inequalities are: Hermite-Hadamard and Fejér.... more

We present the notion of reciprocally (s,m)-convex functions and present some examples and properties of them. We derive some inequalities for this new class of functions, specifically these inequalities are: Hermite-Hadamard and Fejér. In addition, we present some applications of our results to special media of positive real numbers.

The aim of this paper is to describe the method of induction of generalized annotated programs called IGAP what is a special case of inductive fuzzy logic programming for monotonely classified data. This method is based on the multiple... more

The aim of this paper is to describe the method of induction of generalized annotated programs called IGAP what is a special case of inductive fuzzy logic programming for monotonely classified data. This method is based on the multiple use of two valued ILP and the syntactical equivalence of fuzzy logic programs and a restricted class of generalized annotated programs. Finally we compare our method with several fuzzy ILP methods.

Let co X (·) denote the convexification operator on bounded real functions on a convex compact set X. Several necessary and sufficient conditions for the operator co X (·) to preserve continuity and uniformly Lipschitz continuity are... more

Let co X (·) denote the convexification operator on bounded real functions on a convex compact set X. Several necessary and sufficient conditions for the operator co X (·) to preserve continuity and uniformly Lipschitz continuity are established. In the special case of a finite dimensional topological vector space, it is shown that (1) the preservation of continuity is equivalent to the closeness of the set of faces of X and (2) the uniform preservation of Lipschitz continuity is equivalent to X being a polytope.

The first public version of TernAPI (ternary diagrams assessment programming interface) software package for ternary phase diagrams calculation by the convex hull method has been developed. Its reliability and efficiency have been proved... more

The first public version of TernAPI (ternary diagrams assessment programming interface) software package for ternary phase diagrams calculation by the convex hull method has been developed. Its reliability and efficiency have been proved on a large set of systems of different kind: organic liquids and fluids mixtures, water solutions, salts, oxides, metallic alloys. A remarkable advantage of TernAPI is a stable work in the case of isolated miscibility gaps (" islands ") described by a uniform Gibbs energy surface equation. It also contains several improvements of phase diagram calculation algorithm, a new module for x–T diagrams polythermal sections calculation and possibility of optimization of thermodynamic models parameters. NRTL model parameters for the acetic acid–N,N-dimethylformamide–cyclohexene ternary system have been optimized in this work. The specialized language based on Ruby and YAML is used for the description of thermodynamic models of phases.

We obtain, using the Birman-Schwinger method, a series of necessary conditions for the existence of at least one bound state applicable to arbitrary central potentials in the context of nonrelativistic quantum mechanics. These conditions... more

We obtain, using the Birman-Schwinger method, a series of necessary conditions for the existence of at least one bound state applicable to arbitrary central potentials in the context of nonrelativistic quantum mechanics. These conditions yield a monotonic series of lower limits on the "critical" value of the strength of the potential (for which a first bound state appears) which converges to the exact critical strength. We also obtain a sufficient condition for the existence of bound states in a central monotonic potential which yield an upper limit on the critical strength of the potential.

In this note, we discuss the definition of the S1-convexity Phenomenon. We first make use of some results we have attained for KS2 in the past, such as those contained in [1], to refine the definition of the phenomenon. We then observe... more

In this note, we discuss the definition of the S1-convexity Phenomenon. We first make use of some results we have attained for KS2 in the past, such as those contained in [1], to refine the definition of the phenomenon. We then observe that easy counter-examples to the claim K1 extends K0 are found. Finally, we make use of one theorem from [2] and a new theorem that appears to be a supplement to that one to infer that does not properly extend K0 in both its original and its revised version.

This paper contains a new efficient algorithm to construct the convex hull of a set of points in the plane. The proposed algorithm is able to find the points on the convex hull in boundary traversal order. When the convex hull has... more

This paper contains a new efficient algorithm to construct the convex hull of a set of points in the plane. The proposed algorithm is able to find the points on the convex hull in boundary traversal order. When the convex hull has collinear points, the algorithm can detect all the collinear points on the hull without skipping the intermediate points. Furthermore it can deal with the data sets where coincident points appear. Two main methods have been used to make the algorithm efficient. First one is achieving parallelism which is done by partitioning the data set. Second one is data reduction which is done by removing unnecessary points at each step of processing. Further we have proved that the performance of our algorithm is better than interior points algorithm by experimental comparison.

We present an abstract framework for irreversible rate independent evolution processes of quasi-static nature. The main tool relies on the minimizing movement theory. In particular situations, stability and energy inequality of Mielke’s... more

We present an abstract framework for irreversible rate independent evolution processes of quasi-static nature. The main tool relies on the minimizing movement theory. In particular situations, stability and energy inequality of Mielke’s type are satisfied. Several examples are given, among which the obstacle erosion.

Let f: X! IR f+1g be a lower semicontinuous function on a Banach space X. We show that f is quasiconvex if and only if its Clarke subdi erential@ f is quasimonotone. As an immediate consequence, we get that f is convex if and only if@ f... more

Let f: X! IR f+1g be a lower semicontinuous function on a Banach space X. We show that f is quasiconvex if and only if its Clarke subdi erential@ f is quasimonotone. As an immediate consequence, we get that f is convex if and only if@ f is monotone.

In order to fix the ideas, we suppose that X is a real normed space with topological dual X∗. Given an extended-real-valued function Φ : X → R ∪ {+∞}, let us write as usual Dom Φ for the set of all elements x ∈ X for which Φ(x) is finite,... more

In order to fix the ideas, we suppose that X is a real normed space with topological dual X∗. Given an extended-real-valued function Φ : X → R ∪ {+∞}, let us write as usual Dom Φ for the set of all elements x ∈ X for which Φ(x) is finite, and say that Φ is proper if DomΦ 6= ∅. Let Γ0(X) denote the set of all the convex, proper and extended-real-valued functions defined on X and recall that the subdifferential of Φ at x is given by

In this work the concept of φ −convex stochastic process is introduced and certain algebraic properties are deduced. Also, some mean square integral inequalities of Hermite-Hadamard type are established. In addition, various mean square... more

In this work the concept of φ −convex stochastic process is introduced and certain algebraic properties are deduced. Also, some mean square integral inequalities of Hermite-Hadamard type are established. In addition, various mean square integral inequalities are studied in order to find upper estimates with the use of the weighted arithmetic mean, the weighted power mean of order p and the logarithmic mean.

In this note, we present a few more important scientific remarks regarding the S−convexity phenomenon. This time, we talk about examples. That was one of the first queries Professor Mark Nelson had for us at the ANZIAM meeting that... more

In this note, we present a few more important scientific remarks regarding the S−convexity phenomenon. This time, we talk about examples. That was one of the first queries Professor Mark Nelson had for us at the ANZIAM meeting that happened this year, in 2017, at the Wol-longong University, Information Sciences building. We here talk about a very trivial example. Yet this example will prove a few really old results to be equivocated. MSC(2010): 26A51

Every lower semicontinuous convex function can be represented through its subdifferential by means of an "integration" formula introduced in (10) by Rockafellar. We show that in Banach spaces with the Radon-Nikodym property this... more

Every lower semicontinuous convex function can be represented through its subdifferential by means of an "integration" formula introduced in (10) by Rockafellar. We show that in Banach spaces with the Radon-Nikodym property this formula can be significantly refined under a standard coerciv- ity assumption. This yields an interesting application to the convexification of lower semicontinuous functions.