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In previous papers we have studied topical functions f:X-> K and related classes of functions, where X is a b-complete semimodule over an idempotent b-complete semifield K. Without essential restriction of the generality, we assume that K has no greatest element sup K, and hence for x in X and y=inf X the residuation x/y is not defined. Now we adjoin to K an outside "greatest element" top =sup K, and extending in a suitable way the operations of multiplication and addition from K to bar K, which is the union of K and top, we study "extended functions" f:X-> bar K. Actually, we give two different extensions of the product from K to bar K, so as to obtain a meaning also for the residuation x/inf X, with any x in X, and in particular for inf X/inf X, and use them to give characterizations of topical (i.e. increasing homogeneous, defined with the aid of the first product) and anti-topical (i.e. decreasing anti-homogeneous, defined with the aid of the second product) functions f:X-> bar K, by some inequalities. Next we study for functions f:X -> bar K their conjugates and biconjugates of Fenchel-Moreau type with respect to the coupling functions phi(x,y)=x/y x,y in X and psi(x,(y,d)):=inf {x/y,d}; x,y in X,d in bar K, and obtain characterizations of topical and anti-topical functions. In the subsequent sections we consider the polars of a subset G in X for the coupling functions phi and psi, and the support set of a function f:X->K with respect to the set tilde T of all "elementary topical functions" tilde t_{y}(x):=x/y, x in X, y in X-{inf X} and two concepts of support set of f:X-> bar K at a point x_0 in X., 41 pages

We give some contributions to the theory of “max–min convex geometry”, that is, convex geometry in the semimodule R max , min n over the max-min semiring R max , min = R ∪ { - ∞ , + ∞ } . We introduce “elementary segments” that generalize from n = 2 the horizontal, vertical or oblique segments contained in the main bisector of R max , min 2 . We show that every segment in R max , min n is a concatenation of a finite number of elementary subsegments (at most 2 n - 1 , respectively at most 2 n - 2 , in the case of comparable, respectively, incomparable, endpoints x , y ). In this first part we study “max–min segments”, and in the subsequent second part (submitted) we study “max–min semispaces” and some of their relations to “max–min convex sets”.

A max-plus hyperplane (briefly, a hyperplane) is the set of all points x=(x1,…,xn)∈Rmaxn satisfying an equation of the form a1x1⊕⋯⊕anxn⊕an+1=b1x1⊕⋯⊕bnxn⊕bn+1, that is, max(a1+x1,…,an+xn,an+1)=max(b1+x1,…,bn+xn,bn+1), with ai,bi∈Rmax(i=1,…,n+1), where each side contains at least one term, and where ai≠bi for at least one index i. We show that the complements of (max-plus) semispaces at finite points z∈Rn are “building blocks” for the hyperplanes in Rmaxn (recall that a semispace at z is a maximal – with respect to inclusion – max-plus convex subset of Rmaxn⧹{z}). Namely, observing that, up to a permutation of indices, we may write the equation of any hyperplane H in one of the following two forms:a1x1⊕⋯⊕apxp⊕ap+1xp+1⊕⋯⊕aqxq=a1x1⊕⋯⊕apxp⊕aq+1xq+1⊕⋯⊕amxm⊕an+1,where 0⩽p⩽q⩽m⩽n and all ai(i=1,…,m,n+1) are finite, or,a1x1⊕⋯⊕apxp⊕ap+1xp+1⊕⋯⊕aqxq⊕an+1=a1x1⊕⋯⊕apxp⊕aq+1xq+1⊕⋯⊕amxm⊕an+1,where 0⩽p⩽q⩽m⩽n, and all ai(i=1,…,m) are finite (and an+1 is either finite or -∞), we give a formula that expresses a nondegenerate strictly affine hyperplane (i.e., with m=n and an+1>-∞) as a union of complements of semispaces at a point z∈Rn, called the “center” of H, with the boundary of a union of complements of other semispaces at z. Using this formula, we obtain characterizations of nondegenerate strictly affine hyperplanes with empty interior. We give a description of the boundary of a nondegenerate strictly affine hyperplane with the aid of complements of semispaces at its center, and we characterize the cases in which the boundary bd H of a nondegenerate strictly affine hyperplane H is also a hyperplane. Next, we give the relations between nondegenerate strictly affine hyperplanes H, their centers z, and their coefficients ai. In the converse direction we show that any union of complements of semispaces at a point z∈Rn with the boundary of any union of complements of some other semispaces at that point z, is a nondegenerate strictly affine hyperplane. We obtain a formula for the total number of strictly affine hyperplanes. We give complete lists of all strictly affine hyperplanes for the cases n=1 and n=2. We show that each linear hyperplane H in Rmaxn (i.e., with an+1=-∞) can be decomposed as the union of four parts, where each part is easy to describe in terms of complements of semispaces, some of them in a lower dimensional space.

We give some complements to Nitica, V. and Singer, I., 2007, Max-plus convex sets and max-plus semispaces, I.Optimization, 56, 171–205. We show that the theories of max-plus convexity in and -convexity in are equivalent, and we deduce some consequences. We show that max-plus convexity in Rn is a multi-order convexity. We give simpler proofs, using only the definition of max-plus segments, of the results of loc. cit. on max-plus semispaces. We show that unless ≤ is a total order onA, the results ofloc. cit. on semispaces cannot be generalized in a natural way to the framework ofAn =(An , ≤, ⊗), whereA:=M∪ {−∞}, withM=(M,≤,⊗) being a lattice ordered group and −∞ a “least element’ adjoined toM.

We give simpler proofs of some known conjugation formulas and subdifferential formulas of convex analysis and we give some new interconnections between them, showing how each of them follows from the others.

We extend some recent results on radiant and normal sets in , increasing positively homogeneous and increasing co-radiant functions plus-radiant and downward sets in , topical and sub-topical functions to the unifying framework of subsets of An and functions f:An → where A=(A,≤, ⊗ ) is a conditionally complete lattice ordered group and is the minimal enlargement of A. For results involving closedness, continuity and semi-continuity, we assume that the lattice A is continuous and we use the order topology on An and A.

We extend the results of Rubinov and Glover on the separation of normal subsets of , and those of Martinez-Legaz, Rubinov and Singer on the separation of downward subsets of , to the unifying framework of subsets of An , where A is a continuous conditionally complete lattice ordered group. We also extend earlier results of the second author on the characterizations of conjugations of type Lau of functions , to the framework of functions , where is the canonical enlargement of A. These extensions are based on the notions and results of the theory of continuous lattices, generalized in an earlier work of the first author and here, to the conditionally complete case, in particular the way below order relation ≪, and the Scott and Lawson topologies. We also introduce the “way above order relation ≫”, and consider the “bi-Scott topology” on A, introduced previously by the first author, which under our continuity assumption on A, turns out to be equal to the order topology; moreover, we show that for this topol...

The theory of lower semicontinuous linear mappings for dioid-valued functions and the abstract (axiomatic) theory of Fenchel-Moreau conjugations have developed parallelly and independently. In this paper we show that these theories are closely related. We introduce the concept of conjugations for functions with values in the top enlargement of a boundedly complete idempotent semifield. Main result: There exists an isomorphism between the complete semimodules of all lower semicontinuous linear mappings and all conjugations. We show that the above framework and the framework of functions with values in the canonical enlargement of a complete lattice ordered group, used in our previous papers to introduce and study abstract conjugations, are essentially equivalent.

Applying for the additive min-type coupling function $$ \varphi (x, w) = \mathop {\min }\limits_{1 \le i \le n} (x_1 + w_i )\quad (x = (x_i ), w = (w_i ) \in {\shadR}^n ) $$ the techniques from abstract convex analysis, developed in [I. Singer (1997). Abstract Convex Analysis . Wiley-Interscience, New York], we show that the conjugate of type Lau f L ( } ) of f with respect to } has the simple expression $ f^{L(\varphi )} (w) = - f( - w)\, (w \in {\shadR}^n ) $ if and only if f is increasing and upper semi-continuous. As a consequence, we obtain that for topical (i.e., increasing additively homogeneous) functions f the Fenchel-Moreau conjugate of f with respect to } and the conjugate of type Lau of f with respect to } coincide.

We develop a theory of downward subsets of the space \R I, where I is a finite index set. Downward sets arise as the set of all solutions of a system of inequalities x∈\R I, ft(x)≤0 (t∈T), where T is an arbitrary index set and each ft (t∈T) is an increasing function defined on \R I. These sets play an important role in some parts of mathematical economics and game theory. We examine some functions related to a downward set (the distance to this set and the plus-Minkowski gauge of this set, which we introduce here) and study lattices of closed downward sets and of corresponding distance functions. We discuss two kinds of duality for downward sets, based on multiplicative and additive min-type functions, respectively, and corresponding separation properties, and we give some characterizations of best approximations by downward sets. Some links between the multiplicative and additive cases are established.

The theory of lower semicontinuous linear mappings for dioid-valued functions and the abstract (axiomatic) theory of Fenchel-Moreau conjugations have developed parallelly and independently. In this paper we show that these theories are closely related. We introduce the concept of conjugations for functions with values in the top enlargement of a boundedly complete idempotent semifield. Main result: There exists an isomorphism between the complete semimodules of all lower semicontinuous linear mappings and all conjugations. We show that the above framework and the framework of functions with values in the canonical enlargement of a complete lattice ordered group, used in our previous papers to introduce and study abstract conjugations, are essentially equivalent.

For the reverse convex infimization problem (P) inf f(X\G)where G convex subset of a locally convex space X and is a function, we.carry ouexplicitly one of the main ideas of our previous paper [11], namely, we obtain some new Lagrangian duality theorems for (P) from the surrogate duality theorems and of our recen paper [16].

We introduce and study the Abadie constraint qualification, the weak Pshenichnyi--Levin--Valadier property, and related constraint qualifications for semi-infinite systems of convex inequalities and linear inequalities. Our main results are new characterizations of various constraint qualifications in terms of upper semicontinuity of certain multifunctions. Also, we give some applications of constraint qualifications to linear representations of convex inequality systems, to convex Farkas--Minkowski systems, and to formulas for the distance to the solution set. Some of our concepts and results are new even in the particular case of finite inequality systems.

Let Xbe a setW set of extended-real valued functions on Xand 0 an element of Xsuch that w(0) = 0 for all win W. We give a general theory of dual representations, without any extra parameters, of various hulls for extended-real valued functions on X satisfying f(0)= inf f(X\{0})We use the unifying framework of quasi-convex functions with respect to families of subsets of Xand a condition generalizing the bipolar theorem. Our results contain, as particular cases, some recent results of Thach (1991, 1993) and Rubinov and Glover (1996) on W-pseudo-affine and W-quasi-coaffine hulls.We also give some results in the converse direction. These yield, in particular, that the bipolar theorem is equivalent to a certain property of the lower semi-continuous quasi-convex hulls of functions on locally convex spaces

We give duality theorems and dual characterizations of optimal solutions for abstract quasi-convex supremization problems and infimization problems with abstract reverse convex constraint sets. Our main tools are dualities between families of subsets, conjugations of type Lau associated to them, and subdifferentials with respect to conjugations of type Lau. These tools permit us to give explicitly the relation between.the constraint sets, and the relation between the objective functions, of the primal problem and the dual problem. As applications, we obtain duality theorems for quasi-convex supremization and reverse convex infimization in locally convex spaces and, in particular, for worst and best approximation in normed linear spaces.

We give some results on the existence of global error bounds for convex multifunctions between normed linear spaces (until the present, only some results on local error bounds have been known in this general setting). As applications we obtain, among others, improvements of a theorem of Robinson on global error bounds for convex inequalities, of a result of Luo and Tseng on uniform boundedness of the Hoffman constants for linear inequalities and equalities, and of Lotov's result on pointwise Lipschitz continuity of the solution sets of linear inequalities, with respect to data perturbations.

Main result: For each closed convex subset C of a Banach space X and each x ¬∈ C there exists an equivalent norm on X for which x has a unique nearest point in C.

Recently Deutsch, Li and Swetits [2] have studied, in Hilbert space, a dual problem (Qm ) to the primal problem (P) of minimization of a special class of convex functions f over the intersection of m closed convex sets, where m is finite. In the first part of this paper we obtain, in a locally convex space, some results on problem (Qm ) and on its relations with the usual Lagrangian dual problem (Q) to (P) (studied in [9]), in the case when (P) has a solution. In the second part we give some applications to duality for the distance to the intersection of m closed convex sets in a normed linear space, in the case when a nearest point exists. Most of our results seem to be new even in the particular cases studied in [9] (the case m = 1), [l] (duality formulas for the distance to the intersection of m closed half-spaces in a normed linear space) and [2].

Here we want to present briefly some results, problems and directions of research in the modern theory of best approximation, i.e. in which the methods of functional analysis are applied in a consequent manner. In this theory the functions to be approximated and the approximating functions are regarded as elements of certain normed linear (or, more generally, of certain metric) spaces of functions and best approximation amounts to finding “nearest points”. The advantages and a brief history of this modern point of view have been described in the Introduction to the monograph [82] and we shall not repeat them here; the material which will be presented in the sequel will be convincing enough, we hope, to prove again that the theory of best approximation in normed linear spaces constitutes both a rigorous theoretical foundation for the existing classical and more recent results in various concrete spaces and a powerfull tool for obtaining new results, solving the new problems which appear.

We continue our study of generalized conjugations for functions with values in the canonical enlargement of a complete ordered group, started in [10], which encompass various kinds of known conjugations and polarities. We obtain extensions, to this framework, of some results on d.c. duality theory and subdifferentials, and we give some applications to conjugations and subdifferentials for functions with values in \(\bar R_ + = [0, + \infty ] \).

We present some results extending d.c. duality theory to the generalized convex setting.Given a binary operation ∗ on the extended real line, we consider the problem of minimizingƒ∗–hwithƒ and h being functions defined on an arbitrary set. For this problem. we obtain results which encompass, as particular cases, some well known duality theorems for the infimum of the difference of two functions and the infimum of the maximum of two functions, as well as some well known formulae for the conjugate of the difference of two functions and the conjugate of type Lau of the maximum of two functions. Instead of considering the Fenchel conjugation operator, our results are expressed with the aid of dualities which are associated to the operation ∗.We also study. for such dualities, the corresponding generalized subdifferential

Continuing our papers [4] - [7], where we have given an axiomatic approach to unperturbational dual problems, we study here perturbational dual problems. we give some characterizations of various classes of perturbations and the associated marginal functions and of various classes of perturbational dual objective functions and the associated Lagrangians, regarded as functions of their natural variables and of the primal parameters

It is proved that ifX andY are linear spaces andF :X →p(Y) is a set-valued map with convex graph such thatF(x) ≠ O for allx ∈X andF(x 0) is a singleton for somex 0, thenF is single-valued and affine. Applications to metric projections and to adjoints of set-valued maps are given.

For constrained primal infimization problems, we give some characterizations of the surrogate dual objective functions, regarded as functions of three variables (namely, of the primal constraint set, the primal objective function and the dual variable). We also give some results on surrogate constraint mappings

We introduce and study dualities △:[Rbar]x → [Rbar]w (i.e., mappings f e [Rbar]x → f △ e [Rbar]w such that for all {fi }ie1 ∈ Rx and all index sets I), which satisfy the additional condition and their duals, which are characterized as those dualities △*:Rx → Rw for which , where ⊥ and ⊺ are two new binary operations on [Rbar], which we introduce here. Furthermore, we give a characterization of those Δ which are also conjugations. Some applications are also mentioned.

We study dualities between two complete lattices Eand Fi.e., mappings △:E→ F satisfying for all {x i } ieI ⊆E and all index sets I including the empty set I = O. We give characterizations and representations of dualities △, and some results on the dual △* F→Eof △ and on the associated hull operator △*△:E→Ein the general case and in various particular eases. Among several applications, we devote special attention to Fenchel-Moreau conjugations.

We study the lexicographical index of the elements of R n , which we introduce here, and the linear operators on R n which preserve the lexicographical order or the lexicographical index, as well as those which do not increase or do not decrease the lexicographical index.

We show that the (pointwise) supremum of two arbitrary hull operators on a complete lattice is again a hull operator, which can be expressed in a natural way in terms of the two initial hull operators. Our results yield, in particular, very simple proofs of some results of Atteia and ElQortobi on projective bipolars of sets and functions. Also, we give a new way of recuperating abstract convex hulls of sets from abstract quasi-convex hulls of their indicator functions.

Let G be a non-empty subset of a real locally convex space F and h:F→[rbar] a functional. We give necessary and sufficient conditions in order that goeG should satisfy h(go)=inf h(G) and that there should hold one of the formulae of (weak or stronq) duality of [5], [6] for inf h(G). The conditions are similar to those of [5], [6], but involve some additional level sets of h.

We introduce some new abstract Pontryagin maximum principles (PMP's) for linear systems, corresponding to our formulae of Lagrangian duality and surrogate duality of [7] and [10]. §1 is concerned with PMP's for linear systems and their study by passing to F/Ker u. §2 is devoted to the equivalent setting of PMP's involving only one space X and their study via the primal functional corresponding to perturbations by translations.

We show that there exists a conjugate Banach space E = B* with a basis, such that the distance from E to the collection of all conjugate Banach spaces can be made arbitrarily large, by suitable renorming of E. This solves a problem raised by B.V. Godun, Dokl. Akad. Nauk SSSR. 236 (1977), 18–20.

We consider the optimization problem inf h(G),_where G≠O is a subset of a real locally convex space F and h:F→[Rbar].§1 contains a unified approach to the necessary and sufficient conditions of [l3]–[l5] and of §§2–3. In §§2–3 we give conditions for the validity of duality formulae involving half-spaces (closed or open), unions of strips and unions of closed strips, containing G. In §4 we show some relations between the minimization of h and hq (the quasi-convex hull of h) on G and ∞ G, or ∞[bbar] G and we determine . In §5 we consider duality formulae involving nonlinear functionals.

We show that, if (F →u X) is a linear system, Ω ⊂ X a convex target set and h: X → R a convex functional, then, under suitable assumptions, the computation of inf h({y ϵ F ¦ u(y) ϵ Ω} ) can be reduced to the computation of the infimum of h on certain strips or hyperplanes in F, determined by elements of u ∗ (X ∗ ) , or of the infima on F of Lagrangians, involving elements of u ∗ (X ∗ ) . Also, we prove similar results for a convex system (F →u X) and the convex cone Ω of all non-positive elements in X.

XEM xsM (the extension to complex scalars can be obtained with the usual methods-see [3,4]) and therefore, from now on, we shall omit the adjective “real”. In several works it has been observed that Theorem A yields, as a particular case, the value of the intimum of a continuous convex functional f on a convex subset G of E, by taking in (1.1) h = -6,, where 6, is the indicator of the set G, i.e.

On etablit certaines relations entre dualites au sens de Evers et van Maaren (1981), polarites comme celles utilisees dans Griffin, Araoz et Edmonds (1982), fonctionnelles de couplage au sens de Moreau (1966-1967, 1970) et conjugaisons comme celles introduites par l'auteur (1983)