σ-POROSITY IN MONOTONIC ANALYSIS WITH APPLICATIONS TO OPTIMIZATION

We introduce and study some metric spaces of increasing positively homogeneous (IPH) functions, decreasing functions, and conormal (upward) sets. We prove that the complements of the subset of strictly increasing IPH functions, of the subset of strictly decreasing functions, and of the subset of strictly conormal sets are σ-porous in corresponding spaces. Some applications to optimization are given.


Introduction
Consider a normed space X equipped with the order relation ≥ induced by a convex open cone C such that cl C is a pointed cone.(More general conic sets C are considered in some parts of the paper.)We study metric spaces of increasing positively homogeneous of degree one (IPH) functions defined on C, decreasing nonnegative functions defined on C, and conormal (upward) subsets of C. Recall that the set V ⊂ C is called conormal if (x ∈ V , y ≥ x) ⇒ y ∈ V .We show that the complements of the subset of strictly increasing IPH functions, of the subset of strictly decreasing functions, and of the subset of strictly conormal sets are σ-porous in the corresponding spaces.First results of this kind were presented in [10].We use a modification of some constructions from [10] in this paper.
There are some links between IPH functions, decreasing functions, and conormal sets.These links are based on the following observation: consider a set V ⊂ C * := C × (0,+∞).Then the following assertions are equivalent.
(1) V is a closed and conormal set.
(2) V is the upper-level set {x : p(x) ≥ 1} of an IPH function p defined on C * .
(3) V is the epigraph of a lower semicontinuous nonnegative decreasing function g defined on C. Using corresponding bijections we can extend results obtained for metric spaces of IPH functions to metric spaces of conormal sets and decreasing functions.
The main results are obtained for the set ᏼ 0 of IPH functions p such that 0 < inf x∈C, x =1 p(x) ≤ sup x∈C, x =1 p(x) < +∞.The number ρ(p, q) = max sup x∈C p(x) q(x) ,sup x∈C q(x) p(x) (1.1) can be considered as a natural measure for the estimation of the closeness of functions p, q ∈ ᏼ 0 .This observation leads to the introduction of the natural metric d(p, q) = lnρ(p, q) on the set ᏼ 0 .It is easy to see that the metric space (ᏼ 0 ,d) is complete.Note that d(p, q) does not depend on the "size" of the pair p, q in the sense that d(p, q) = d(λp,λq) for all λ > 0. This number describes the distance in terms of the "shape" of this pair.Using above-mentioned bijections we introduce metric spaces of conormal sets and decreasing nonnegative functions and study their properties.
We also discuss applications of the results obtained to the examination of some questions arising in optimization.Consider the set H of optimization problems P( f ,g): minimize f (x) subject to g(x) ≤ 0, ( where f ,g : R m → R are functions with some properties.Many questions related to problem P(f ,g) can be expressed in terms of the perturbation function β f ,g (y)=inf x:g(x)≤y f (x).This function is decreasing.We say that problems P( f ,g) ∈ H and P( f ,g ) ∈ H are equivalent if β f ,g = β f ,g .Thus we can include the set Ᏼ of classes of equivalent pairs of optimization problems in the metric space of decreasing functions.Using results obtained, we can prove that the subset of Ᏼ that consists of pairs π such that at least one problem from π has no solutions on the boundary of the set of feasible elements, is σporous in the metric space under consideration.

Preliminaries
A set C in a normed space X is called conic if λx ∈ C for all x ∈ C and λ > 0. A convex conic set is called a convex cone.A convex cone K generates the order relation ≥ on X, namely x ≥ y if x − y ∈ K.We will write x > y if x ≥ y and x = y.Let C be a conic set.A function p : C → R +∞ is called positively homogeneous of degree one (PH) if p(λx) = λp(x) for all λ > 0. Denote by Ꮾ the intersection of C and the unit ball and by the intersection of C and the unit sphere: Clearly each IPH function is completely defined by its trace on : if p 1 , p 2 are PH functions and p 1 (x) = p 2 (x) for all x ∈ , then p 1 (x) = p 2 (x) for all x ∈ C. Let p : C → R be a finite PH function.The quantity is called the norm of p.The following simple assertion is well known and can be easily proved.
Proposition 2.1.A finite PH function p defined on a conic set C is continuous at zero if and only if p < +∞.
Let C be a conic set in a normed space X with the order relation ≥ induced by a closed convex pointed cone

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A function p : C → R is called strictly increasing if x > y⇒p(x) > p(y).In this paper, we will study increasing and strictly increasing PH functions.We will use the abbreviation IPH for increasing PH functions.Since the ordering convex cone K contains C, it follows that x ≥ 0 for each x ∈ C.
The following proposition holds.A proof of Proposition 2.2 for the case C = K can be found in [3,Proposition 6].This proof is valid in the general case and we omit it.
We denote by ᏼ * the set of all nonnegative PH functions defined on C and continuous at zero.We assume that ᏼ * is equipped with a metric d u defined by (2.3) Clearly the metric space (ᏼ * ,d u ) is complete.

Metric spaces of equivalent IPH functions
Let C be a conic set in a normed space X.Consider a conic set ᏼ ⊂ ᏼ * of continuous at zero nonnegative PH functions p : C → R. We assume that ᏼ is closed with respect to uniform convergence on = {x ∈ C : x = 1}.We give some examples of a set ᏼ: the set of nonnegative and continuous at zero PH functions, the set of nonnegative continuous PH functions, the set of continuous at zero IPH functions (it is assumed that there is an order relation introduced by a closed convex pointed cone K ⊃ C).We say that functions p, q ∈ ᏼ are equivalent and write p ∼ q if there exist constants 0 < γ ≤ Γ < +∞ such that Clearly ∼ is an equivalency relation.Let r ∈ ᏼ.Consider the class ᏼ r of functions equivalent to a function r.It easily follows from the definition of the equivalence that ᏼ r is a conic set.Assume in the sequel that 0/0 = 0. Then the ratio p(x)/q(x) is finite for all It is easy to check that the function ρ possesses the following properties: ρ(p, q) = ρ(q, p) ∀p, q ∈ ᏼ r ; (3.4) ρ(p, q) ≥ 1 ∀p, q ∈ ᏼ r ; (3.5) ρ(p, q) = 1 iff p = q; (3.6) ρ(p,h) ≤ ρ(p, q)ρ(q,h) ∀p, q,h ∈ ᏼ r ; (3.7) ρ(λp,λq) = ρ(p, q) ∀p, q ∈ ᏼ r , λ > 0. (3.8) We only check (3.7).Let p, q ∈ ᏼ r .Then we have for all (3.9) These inequalities imply (3.7).It follows from (3.4)-(3.7)that the function is a metric on ᏼ r .Note that d is positively homogeneous of degree zero: d(λp,λq) = d(p, q) for all p, q ∈ ᏼ r and λ > 0. This follows from (3.8).
It follows from Proposition 3.1 that a sequence p n converges to p in the metric space (ᏼ r ,d) if and only if there exists a sequence δ n > 0, δ n → 0 such that Consider the set ᏼ 0 of functions p ∈ ᏼ such that inf{p(x) : x ∈ } > 0. Clearly each two functions p, q ∈ ᏼ 0 are equivalent and ᏼ 0 coincides with ᏼ r for arbitrary r ∈ ᏼ 0 .The following assertion demonstrates that the convergence in the space (ᏼ 0 ,d) follows from uniform convergence, therefore (see Proposition 3.2) coincides with the uniform convergence.
Proposition 3.3.Let p n ∈ ᏼ 0 , n = 1,... and let p ∈ ᏼ 0 .If the sequence p n converges to p uniformly on the set , then p n converges to p in the metric space (ᏼ 0 ,d).
Then for all x ∈ and large enough n we have

Porosity results for metric spaces of IPH functions
In this section, we consider a closed subspace ᏼ of the metric space (ᏼ * ,d u ) (see Section 2 for the definition of this space) that consists of IPH functions.In particular, it can be the set of continuous at zero IPH functions or continuous everywhere IPH functions.Since ᏼ is closed, it follows that the metric space (ᏼ,d u ) is complete.We will show that under some natural assumptions the complement of the set of strictly increasing PH functions from ᏼ is σ-porous in the space (ᏼ,d u ).First we give the definition of porosity and σ-porosity, which will be used in the paper.There are various definitions of porocity (see [6] for references and a discussion).We use the following definition (see, e.g., [1,2,5,10]).Let (P,d) be a complete metric space.Denote the ball {q ∈ P : d(p, q) ≤ r} by B(p,r).A set Ω ⊂ X is called porous in (P,d) if there exist α ∈ (0,1] and r 0 > 0 such that for each positive r < r 0 and each p ∈ P there exists a ball B( p,αr) of radius αr such that B( p,αr) ⊂ B(p,r) and B( p,αr) ∩ Ω is empty.Later on we consider a set Ω that is the complement to a set P .Then P \ P is porous if there exist α > 0 and r 0 > 0 such that for each r ∈ (0,r 0 ) and for each p ∈ P an element p can be found for which B( p,αr) ⊂ B(p,r) ∩ P .
A set Ω is called σ-porous in (P,d) if Ω is a countable union of porous sets.
Assume that the set ᏼ contains a strictly increasing function l such that If ᏼ is a convex cone, then the property (4.1) is valid.We fix such a function l and assume without loss of generality that l = 1.Note that the function p + λl from (4.1) is strictly increasing for each p ∈ ᏼ and each λ > 0. The following definition is an extension of the definition given in [10].
Definition 4.1.A function p ∈ ᏼ is called strictly increasing with respect to l if for any positive integer n there exists a δ n > 0 such that It is easy to check that each strictly increasing with respect to l function is strictly increasing.Indeed, let x < y.Since l is strictly increasing, it follows that there exists n such that l(x) < l(y) − 1/n.Then p(x) < p(y) − δ n , hence p(x) < p(y).Denote by ᏼ l the set of all strictly increasing with respect to l functions.
Proof.For each positive integer n consider the set ᏼ l n of all IPH functions p ∈ ᏼ such that there exists δ n > 0 with the property (4.2).It follows from the definition of ᏼ l that ᏼ l = ∞ n=1 ᏼ l n , so Thus we need to prove that the set ᏼ \ ᏼ l n is porous in ᏼ.Let n be a natural number.Consider a number r ∈ (0,1] and numbers α and γ such that We have Clearly p ∈ ᏼ.Since l is strictly increasing, it follows that p is also strictly increasing.We now show that p ∈ ᏼ l n .Indeed, let x, y ∈ Ꮾ, x ≤ y, and l(x) < l(y) − 1/n.Then This means that p ∈ ᏼ l n with δ n = γ/n.Since l = 1, it follows that d(p, p) = γ.Let q ∈ ᏼ and d( p, q) < αr.Using (4.5) we have so B( p,αr) ⊂ B(p,r).We now show that q ∈ ᏼ l n .Let x, y ∈ Ꮾ, x ≤ y, and l(x) < l(y) − 1/n.Since d( p, q) < αr, it follows that q(y) ≥ p(y) − αr and −q(x) ≥ − p(x) − αr.Using these inequalities and (4.7) we have Let δ n = γ/n − 2αr.It follows from (4.5) that δ n > 0. Due to (4.9) we have q ∈ ᏼ l n .We have proved that B( p,αr) ⊂ ᏼ l n .This means that ᏼ \ ᏼ l n is a porous set.Remark 4.3.Porosity results for some metric spaces of increasing functions were established in [10].The set of IPH functions is a subset of some of these metric spaces, however a porosity result for a whole space does not imply a similar result for its subspaces.Also spaces with uniform metric were not considered in [10].
We now turn to the space (ᏼ 0 ,d) where ᏼ 0 is the set of functions from ᏼ with the property inf x∈ p(x) > 0 and d is a metric defined by (3.10).We need the following assertion.
Assume that the set ᏼ 0 contains a strictly increasing function l such that (4.1) holds.We fix this function and assume without loss of generality that l = 1.Denote by ᏼ l 0 the set of all strictly increasing with respect to l functions from ᏼ 0 (see Definition 4.1).
Theorem 4.5.The set ᏼ 0 \ ᏼ l 0 is σ-porous in (ᏼ 0 ,d).Proof.For each natural n consider the set (ᏼ 0 ) l n of all functions p ∈ ᏼ 0 such that there exists δ n > 0 with the property (4.2).We have (4.16) For each natural m consider the set Due to (4.16) we have Thus, in order to obtain the result, we need to prove that the set Q m,n := Q m \ (ᏼ 0 ) l n is porous for each of the positive integers n and m.We fix natural m and n.Without loss of generality assume that m ≥ 2. Let M = m(4m + 1)n.Due to Lemma 4.4, we can find numbers α > 0 and r 0 > 0 such that for each r ∈ (0,r 0 ) there exists γ 1 for which (4.10) and (4.11) hold.We can assume without loss of generality that r 0 ≤ 1.We fix a number r ∈ (0,r 0 ) and consider a number γ 1 corresponding to r.We also need the number Note that γ < 1. Indeed due to (4.10), we have γ We check that B( p,αr) ⊂ (ᏼ 0 ) l n .Let q ∈ B( p,αr) and let x, y ∈ Ꮾ be vectors such that x ≤ y and l(x) < l(y) − 1/n.Due to Proposition 3.1 and the inequality d( p, q) < αr, we have p(y) − q(y) ≤ e αr − 1 q(y), q(x) − p(x) ≤ e αr − 1 p(x), (4.29) hence q(y) − q(x) ≥ p(y) − p(x) − e αr − 1 p(x) + q(y) .It follows from (4.31) that q ∈ (ᏼ 0 ) l n .Thus B( p,αr) ⊂ (ᏼ 0 ) l n .We have also proved that B( p,αr) ⊂ B(p,r).This means that the set Q m,n is porous.

IPH functions and conormal sets
Let X be a normed space with the norm • .Let C ⊂ X be a convex pointed cone with the nonempty interior intC.Assume that X is equipped with the order relation ≥ generated by the closed convex cone cl C and the norm • is semimonotone with respect to this order.The latter means that there exists m > 0 such that x ≥ m y if x ≥ y ≥ 0. Semimonotonicity of the norm is equivalent to the normality of the cone cl C (see, e.g., [4]).We fix a point y ∈ intC.If the norm • is semimonotone, then (see, e.g., [4]) the norm • is equivalent to the following norm • y : (5.1) Clearly the norm Note that the unit ball {x : x y ≤ 1} coincides with the set {x ∈ X : −y ≤ x ≤ y}.Assume without loss of generality that • = • y .It follows from this assumption that X is equipped with a monotone norm.We have (5.2) If C is open, then Ꮾ = {x : 0 x ≤ y} where x y means that x − y ∈ intC.We need the following proposition.
Proposition 5.1.Let p : C → R + be an IPH function.Then p is continuous on intC.
Proof.Let x k → x ∈ intC.Since x ∈ intC, it follows that there exists a number t such that tx ≥ y, where y ∈ intC is a reference point which serves for the definition of the norm • = • y .Let ε > 0 be a number such that 1 − εt > 0. Then for large enough k, we have Since p is IPH, we have (5.4) Thus the result follows.
We consider the cone C as a topological space equipped with the natural topology of a subspace.Denote the closure, interior and boundary of a set (5.5)

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We assume that the supremum of the empty set is equal to 0. The Minkowski cogauge can be defined for an arbitrary coradiant set V .(Recall that V is coradiant if x ∈ V ⇒λx ∈ V for all λ ≥ 1.) A presentation of theory of Minkowski cogauges for coradiant sets and in a finite-dimensional space can be found in [7].Many results from [7] can be easily generalized to an arbitrary normed space.In particular, if V is closed (in the topological space C), then Let V be a coradiant set.Then ν V (x) = 0 if and only if ray It is easy to check that the Minkowski cogauge ν V of a coradiant set V is a positively homogeneous of degree one function. ( Since X is equipped with a monotone norm, it follows that the set ᐂ is conormal.An easy calculation shows that ν ᐂ (x) = x for all x ∈ C. Let V be a closed (in the topological space intC) coradiant set.Since the level sets are closed for all r > 0 and the level set {x ∈ C : for a coradiant set V .Indeed, we have for x ∈ C, (5.10) p such that inf x∈ p(x) > 0. Then the restriction of ψ to ᏺ 0 is a bijection between ᏺ 0 and ᏼ 0 .Using this restriction we can introduce a metric d on the set ᏺ 0 , where ) is defined by (3.10).Note the metric space (ᏺ 0 ,d) is complete.

Conormal sets and decreasing functions
Let C be a convex cone in a normed space X such that int C = ∅ and the cone cl C is pointed.Assume that X is equipped with the order relation ≥ generated by cl C and with the norm Consequently, V coincides with the epigraph epig V of the lower cover g V , and g V is an lsc (in the topological space C) and decreasing function.

σ-porosity in monotonic analysis
In the rest of this section we assume that C is an open cone, so C = intC.Then also C * = intC * .
Denote by Ᏸ the set of lower semicontinuous decreasing functions g defined on C and such that dom g is not empty.Let ᏺ * be the totality of conormal closed (in C * ) subsets V of C * such that V = Ꮿ * .Let χ : Ᏸ → ᏺ * be the mapping defined by χ(g) = epig.(6.5)Then χ is a bijection between Ᏸ and ᏺ * .Using this bijection we can define the metric d u on the set Ᏸ: We now consider the set Ᏸ 0 ⊂ Ᏸ that consists of functions g ∈ Ᏸ such that dom g is a bounded set and lim x→0 g(x) < +∞.(Note that lim x→0 g(x) exists since g is a decreasing function defined on C = intC.)Proposition 6.2.g ∈ Ᏸ 0 if and only if there exists m > 0 such that epig ⊃ mᐂ * , where Proof.Let g ∈ Ᏸ 0 .Since m 1 := lim x→0 g(x) = sup x∈C g(x) < +∞, it follows that g(x) < +∞ for all x ∈ C, hence domg = {x ∈ C : g(x) > 0}.Due to the definition of the norm • := • y (see (5.1)), it follows that dom g is bounded if and only if there exists m 2 > 0 such that x ≤ m 2 y for all x ∈ dom g.We also have g(x) ≤ m 1 for all x ∈ dom g.Let y * = (y,1).Since the space X * is equipped with the norm • y * (see (6.2)), it follows that there exists a number m > 0 such that x ∈ dom g ⇒ (x,g(x)) < m.We now show that epig ⊃ mᐂ * .Let (x,λ) ≥ m.First assume that x / ∈ dom g.Then (x,λ) ∈ epig for all λ > 0. It follows from this that if (x,λ) ∈ mᐂ, then (x,λ) ∈ epig.Let x ∈ dom g, then g(x) < m.Let (x,λ) ∈ mᐂ.Then (x,λ) ≥ m.Since x < m, it follows that λ ≥ m > g(x), so (x,λ) ∈ epig.
Assume now that epi g ⊃ mᐂ * with some m > 0. Then the graph {(x, g(x)) : x ∈ dom g} of the function g is placed in the set m Ꮾ with some m > m.It follows from this that g ∈ D 0 .
The restriction of the mapping χ on Ᏸ 0 is a bijection between Ᏸ 0 and the set ᏺ * ,0 of all subsets from ᏺ * that contain mᐂ * with some m.Using this bijection we can introduce the metric d on the set Ᏸ 0 : where ξ 1 (h,g), ξ 2 (h,g) are defined by (6.12) and (6.13), respectively.

Strictly conormal sets and strictly decreasing functions
Let C be an open convex cone in a Banach space X.Assume that the cone clC is pointed and X is equipped with the order relation ≥ generated by cl C. Assume that the norm in X coincides with the norm Proof.(1) Let ν V be strictly increasing.Due to Proposition 5.1, ν V is continuous on the open cone C. We also have V = {x ∈ C : ν V (x) ≥ 1}.It easily follows from this that bd C V = {x : ν V (x) = 1}.Let x ∈ bd C V .If z < x, then ν V (z) < ν V (x) = 1, hence z / ∈ V .We proved that V is strictly normal.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Proposition 3 . 4 .
.16) Since p and p n are positively homogeneous, it follows that (3.16) holds for all x ∈ C. The metric space (ᏼ 0 ,d) is complete.Proof.Let p n ∈ ᏼ 0 be a sequence such that for each ε > 0 there exists N with the following property: for all n > N and all natural m we have d(x n ,x n+m ) < ε.It follows from Proposition 3.1 that for all x ∈ , we have p n (x) < (1 + δ)p n+m (x) and p n+m (x) < (1 + δ)p n (x) with δ = e ε − 1.These inequalities imply inf x∈ p n+m (x) ≥ 1/(1 + δ)inf x∈ p n (x) and sup x∈ p n+m (x) < (1 + δ)sup x∈ p n (x), m = 1,....It is clear that for each x ∈ C there exists p(x) = lim n p n (x) and p ∈ ᏼ 0 .