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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.

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.

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

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.

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.

As natural generalizations of complemented subspaces, we introduce pseudo-complemented and strongly pseudo-complemented subspaces of Banach spaces, and we study for such subspaces some of the problems analogous to the classical problems on complemented and quasi-complemented subspaces.

For a locally convex space E and f :E→R¯, we introduce and study surrogate conjugate functionals of f, which en¬compass the quasi-conjugates [1] , pseudo-conjugates [2] and semi-conjugates [3] of f. Also, we introduce and study surro¬gate convexity of sets GcE and of functionals f:E→R¯ and show their connections with surrogate conjugation and with W-conve-xity of sets [4] and of functionals [5], where WcR¯E. We outline some further developments (surrogate conjugates at a point, surrogate subdifferentials) and an application to optimization. A basic role is played by the concept of a universally defined multifunction A:RXE*→2E

Using our theorems (of [12]) on separation of convex sets by linear operators, in the sense of the lexi-cographical order on Rn, we prove some theorems of surrogate duality for vector optimization problems with convex constraints (but no regularity assumption), where the surrogate constraint sets are generalized half-spaces and the surrogate multipliers are linear operators, or isomorphisms, or isometries. In the cae of inequality constraints, we prove that the surrogate multipliers can be taken lexicographically non-negative isometries or non-negative (in the usual order) linear isomorphisms.

We give some Lagrangian duality theorems for optimization problems, in terms of non-linear separation properties involving epigraphs or related sets. Some applications are also given.

We give a generalization of the notion of the norm of a linear functional and some applications of this generalization.

A complete characterization is given of those functions in C¦a, b¦ which have a unique best approximation from the subspace of spline functions of degree n with k fixed knots. Also, the relationship between unique and strongly unique best approximations from arbitrary finitedimensional subspaces of C0(T) is investigated.

We characterize the total preorders on Rn which are compatible with the vector space structure, in terms of linear operators and the lexicographical order. In particular, we obtain that the lexicographical order is, up to a linear isometry, the unique compatible total order on Rn. We also study compatible total extensions of compatible preorders and give some Szpilrajn type results for compatible preorders and compatible orders.

We extend some results of Rubinov and Singer [13] and Rubinov et al. [14] on topical functions on Rn and downward subsets of Rn to the case where Rn is replaced by a b-complete semimodule X over a b-complete idempotent semifield K. To this end, using residuation, we introduce “elementary” topical functions on X and we show that they are the “building blocks” of the set of all topical functions on X.

We study hemispaces (i,e., convex sets with convex complements) in R n . We give several geometric characterizations of hemispaces and several ways of representing them with the aid of linear operators and lexicographical order. We obtain a metric-affine classification of hemispaces, in terms of their “rank” and “type,” and a “decomposition theorem.” We also give some characterizations of affine transformations which preserve a hemispace.

The results of R. C. James on characterizations of reflexivity of Banach spaces with an unconditional basis in terms of c 0 {c_0} and l 1 {l^1} are extended to arbitrary Banach spaces. Some consequences are obtained.

In the last years the theory of bases in Banach spaces has rapidly developped*). We shall make here some remarks on a few recent results and raise some new problems in this field. The paper has partially an expository character, since we want to make it accessible also to those who are not assumed to know basis theory.

An important particular case of best approximation in normed linear spaces E by elements of linear subspaces G is that when G is closed and the codimension of G (complex or real, according to the space E) is finite: codim G = dim E/G = = n < ∞. Naturally, the results of Chap. I are applicable, in particular, also in the case of best approximation by elements of closed linear subspaces G of finite codimension. However, by using the hypothesis codim G = n, one can obtain additional results, which we shall present in this Chapter.