Earlier and Recent Results on Convex Mappings and Convex Optimization

Keywords: Lower Bounds for Convex Operators, Bounded Finite Dimensional Convex Subsets, Minimum-Norm Elements, Distanced Convex Subsets, Optimization Related To A Markov Moment Problem

Abstract

The main purpose of this review-paper is to recall and partially prove earlier, as well as recent results on convex optimization, published by the author in the last decades. Examples are given along the article. Some of these results have been published recently. Most of theorems have a clear geometric meaning. Minimum norm elements are characterized in normed vector spaces framework. Distanced convex subsets and related parallel hyperplanes preserving the distance are also discussed. The convex involved objective-mappings are real valued or take values in an order-complete vector lattice. On the other side, an optimization problem related to Markov moment problem is solved in the end.

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References

R. B. Holmes, Geometric Functional Analysis and its Applications, Springer, 1975

R.R. Phelps, Lectures on Choquet’s Theorem, D. van Nostrand Company, Inc. Princeton, 1966.

N. Boboc, Gh. Bucur, Convex cones of continuous functions on compact spaces, Academiei, Bucharest, 1976 (Romanian).

H.H. Schaefer, Topological Vector Spaces. Third Printing Corrected, Springer – Verlag, 1971

W. Rudin, Real and complex analysis. Third Ed., McGraw-Hill, Singapore, 1987

R.T. Rockafellar, Convex Analysis,. Theta, Bucharest, 2002 (Romanian)

R. Cristescu, Functional Analysis, Didactical and Pedagogical Publishing House, Bucharest, 1970 (Romanian)

R. Cristescu, Ordered Vector Spaces and Linear Operators, Academiei, Bucharest, Romania, and Abacus Press, Tunbridge Wells, Kent, England, 1976

C. Berg, J.P.R. Christensen, P. Ressel, Harmonic Analysis on Semigroups. Theory of Positive Definite and Related Functions, Springer-Verlag, New York Berlin Heidelberg Tokyo, 1984

C. Niculescu, N. Popa, Elements of Theory of Banach Spaces, Academiei, Bucharest, 1981 (Romanian)

C.P. Niculescu, L.-E. Persson, Convex Functions and Their Applications, Springer-Verlag, 2006

C. Udrişte, Convex Functions and Optimization Methods on Riemannian Manifolds, Springer, Dordrecht, 1994

H. Bonnel, J. Collonge, Optimization over the Pareto outcome set associated with a convex bi-objective optimization problem: theoretical results, deterministic algorithm and application to the stochastic case, Journal of Global Optimization, 62(3) (2015), 481-505

H. Bonnel, L. Tadjihounde, C. Udrişte, Semivectorial Bilevel Optimization on Riemannian Manifolds, Journal of Optimization Theory and Applications, 167(2) (2015), 464-486

C. Drăguşin, Min-max pour des critères multiples, RAIRO Recherche Opérationnelle/Operations Research, 12, 2 (1978), 169-180

D. T. Norris, Optimal Solutions to the L_∞ Moment Problem with Lattice Bounds, PhD Thesis (2002),

University of Colorado, Mathematics Department, 2003

G. Cassier, Problèmes des moments sur un compact de R^n et décomposition des polynȏmes à plusieurs variables, Journal of Functional Analysis, 58 (1984), 254-266

O. Olteanu, Convexité et prolongement d’opérateurs linéaires, C. R. Acad. Sci. Paris, Série A, 286 (1978), 511-514

O. Olteanu, Sur les fonctions convexes définies sur les ensembles convexes bornés de R^n, C. R. Acad. Sci. Paris, Série A, 290 (1980), 837-838

O. Olteanu, Théorèmes de prolongement d’opérateurs linéaires, Rev. Roumaine Math. Pures Appl., 28, 10 (1983), 953-983

O. Olteanu, Application des théorèmes de prolongement d’opérateurs linéaires au problème des moments et à une généralisation d’un théorème de Mazur-Orlicz, C. R. Acad. Sci. Paris, Série I, 313(1991), 739-742

O. Olteanu, A strong separation theorem in normed linear spaces, Mathematica (Cluj), 35(58), 1 (1993), 59-63

O. Olteanu, New results on Markov moment problem, International Journal of Analysis, Vol. 2013, Article ID 901318, pp. 1-17. http://dx.doi.org/10.1155/2013/901318

O. Olteanu, J.M. Mihăilă, Extension and decomposition of linear operators dominated by continuous increasing sublinear operators, U.P.B. Sci. Bull. Series A, 80, 3 (2018), 133-144

R. B. Holmes, Geometric Functional Analysis and its Applications, Springer, 1975

R.R. Phelps, Lectures on Choquet’s Theorem, D. van Nostrand Company, Inc. Princeton, 1966.

N. Boboc, Gh. Bucur, Convex cones of continuous functions on compact spaces, Academiei, Bucharest, 1976 (Romanian).

H.H. Schaefer, Topological Vector Spaces. Third Printing Corrected, Springer – Verlag, 1971

W. Rudin, Real and complex analysis. Third Ed., McGraw-Hill, Singapore, 1987

R.T. Rockafellar, Convex Analysis,. Theta, Bucharest, 2002 (Romanian)

R. Cristescu, Functional Analysis, Didactical and Pedagogical Publishing House, Bucharest, 1970 (Romanian)

R. Cristescu, Ordered Vector Spaces and Linear Operators, Academiei, Bucharest, Romania, and Abacus Press, Tunbridge Wells, Kent, England, 1976

C. Berg, J.P.R. Christensen, P. Ressel, Harmonic Analysis on Semigroups. Theory of Positive Definite and Related Functions, Springer-Verlag, New York Berlin Heidelberg Tokyo, 1984

C. Niculescu, N. Popa, Elements of Theory of Banach Spaces, Academiei, Bucharest, 1981 (Romanian)

C.P. Niculescu, L.-E. Persson, Convex Functions and Their Applications, Springer-Verlag, 2006

C. Udrişte, Convex Functions and Optimization Methods on Riemannian Manifolds, Springer, Dordrecht, 1994

H. Bonnel, J. Collonge, Optimization over the Pareto outcome set associated with a convex bi-objective optimization problem: theoretical results, deterministic algorithm and application to the stochastic case, Journal of Global Optimization, 62(3) (2015), 481-505

H. Bonnel, L. Tadjihounde, C. Udrişte, Semivectorial Bilevel Optimization on Riemannian Manifolds, Journal of Optimization Theory and Applications, 167(2) (2015), 464-486

C. Drăguşin, Min-max pour des critères multiples, RAIRO Recherche Opérationnelle/Operations Research, 12, 2 (1978), 169-180

D. T. Norris, Optimal Solutions to the L_∞ Moment Problem with Lattice Bounds, PhD Thesis (2002),

University of Colorado, Mathematics Department, 2003

G. Cassier, Problèmes des moments sur un compact de R^n et décomposition des polynȏmes à plusieurs variables, Journal of Functional Analysis, 58 (1984), 254-266

O. Olteanu, Convexité et prolongement d’opérateurs linéaires, C. R. Acad. Sci. Paris, Série A, 286 (1978), 511-514

O. Olteanu, Sur les fonctions convexes définies sur les ensembles convexes bornés de R^n, C. R. Acad. Sci. Paris, Série A, 290 (1980), 837-838

O. Olteanu, Théorèmes de prolongement d’opérateurs linéaires, Rev. Roumaine Math. Pures Appl., 28, 10 (1983), 953-983

O. Olteanu, Application des théorèmes de prolongement d’opérateurs linéaires au problème des moments et à une généralisation d’un théorème de Mazur-Orlicz, C. R. Acad. Sci. Paris, Série I, 313(1991), 739-742

O. Olteanu, A strong separation theorem in normed linear spaces, Mathematica (Cluj), 35(58), 1 (1993), 59-63

O. Olteanu, New results on Markov moment problem, International Journal of Analysis, Vol. 2013, Article ID 901318, pp. 1-17. http://dx.doi.org/10.1155/2013/901318

O. Olteanu, J.M. Mihăilă, Extension and decomposition of linear operators dominated by continuous increasing sublinear operators, U.P.B. Sci. Bull. Series A, 80, 3 (2018), 133-144

R. B. Holmes, Geometric Functional Analysis and its Applications, Springer, 1975

R.R. Phelps, Lectures on Choquet’s Theorem, D. van Nostrand Company, Inc. Princeton, 1966.

N. Boboc, Gh. Bucur, Convex cones of continuous functions on compact spaces, Academiei, Bucharest, 1976 (Romanian).

H.H. Schaefer, Topological Vector Spaces. Third Printing Corrected, Springer – Verlag, 1971

W. Rudin, Real and complex analysis. Third Ed., McGraw-Hill, Singapore, 1987

R.T. Rockafellar, Convex Analysis,. Theta, Bucharest, 2002 (Romanian)

R. Cristescu, Functional Analysis, Didactical and Pedagogical Publishing House, Bucharest, 1970 (Romanian)

R. Cristescu, Ordered Vector Spaces and Linear Operators, Academiei, Bucharest, Romania, and Abacus Press, Tunbridge Wells, Kent, England, 1976

C. Berg, J.P.R. Christensen, P. Ressel, Harmonic Analysis on Semigroups. Theory of Positive Definite and Related Functions, Springer-Verlag, New York Berlin Heidelberg Tokyo, 1984

C. Niculescu, N. Popa, Elements of Theory of Banach Spaces, Academiei, Bucharest, 1981 (Romanian)

C.P. Niculescu, L.-E. Persson, Convex Functions and Their Applications, Springer-Verlag, 2006

C. Udrişte, Convex Functions and Optimization Methods on Riemannian Manifolds, Springer, Dordrecht, 1994

H. Bonnel, J. Collonge, Optimization over the Pareto outcome set associated with a convex bi-objective optimization problem: theoretical results, deterministic algorithm and application to the stochastic case, Journal of Global Optimization, 62(3) (2015), 481-505

H. Bonnel, L. Tadjihounde, C. Udrişte, Semivectorial Bilevel Optimization on Riemannian Manifolds, Journal of Optimization Theory and Applications, 167(2) (2015), 464-486

C. Drăguşin, Min-max pour des critères multiples, RAIRO Recherche Opérationnelle/Operations Research, 12, 2 (1978), 169-180

D. T. Norris, Optimal Solutions to the L_∞ Moment Problem with Lattice Bounds, PhD Thesis (2002),

University of Colorado, Mathematics Department, 2003

G. Cassier, Problèmes des moments sur un compact de R^n et décomposition des polynȏmes à plusieurs variables, Journal of Functional Analysis, 58 (1984), 254-266

O. Olteanu, Convexité et prolongement d’opérateurs linéaires, C. R. Acad. Sci. Paris, Série A, 286 (1978), 511-514

O. Olteanu, Sur les fonctions convexes définies sur les ensembles convexes bornés de R^n, C. R. Acad. Sci. Paris, Série A, 290 (1980), 837-838

O. Olteanu, Théorèmes de prolongement d’opérateurs linéaires, Rev. Roumaine Math. Pures Appl., 28, 10 (1983), 953-983

O. Olteanu, Application des théorèmes de prolongement d’opérateurs linéaires au problème des moments et à une généralisation d’un théorème de Mazur-Orlicz, C. R. Acad. Sci. Paris, Série I, 313(1991), 739-742

O. Olteanu, A strong separation theorem in normed linear spaces, Mathematica (Cluj), 35(58), 1 (1993), 59-63

O. Olteanu, New results on Markov moment problem, International Journal of Analysis, Vol. 2013, Article ID 901318, pp. 1-17. http://dx.doi.org/10.1155/2013/901318

O. Olteanu, J.M. Mihăilă, Extension and decomposition of linear operators dominated by continuous increasing sublinear operators, U.P.B. Sci. Bull. Series A, 80, 3 (2018), 133-144

R. B. Holmes, Geometric Functional Analysis and its Applications, Springer, 1975

R.R. Phelps, Lectures on Choquet’s Theorem, D. van Nostrand Company, Inc. Princeton, 1966.

N. Boboc, Gh. Bucur, Convex cones of continuous functions on compact spaces, Academiei, Bucharest, 1976 (Romanian).

H.H. Schaefer, Topological Vector Spaces. Third Printing Corrected, Springer – Verlag, 1971

W. Rudin, Real and complex analysis. Third Ed., McGraw-Hill, Singapore, 1987

R.T. Rockafellar, Convex Analysis,. Theta, Bucharest, 2002 (Romanian)

R. Cristescu, Functional Analysis, Didactical and Pedagogical Publishing House, Bucharest, 1970 (Romanian)

R. Cristescu, Ordered Vector Spaces and Linear Operators, Academiei, Bucharest, Romania, and Abacus Press, Tunbridge Wells, Kent, England, 1976

C. Berg, J.P.R. Christensen, P. Ressel, Harmonic Analysis on Semigroups. Theory of Positive Definite and Related Functions, Springer-Verlag, New York Berlin Heidelberg Tokyo, 1984

C. Niculescu, N. Popa, Elements of Theory of Banach Spaces, Academiei, Bucharest, 1981 (Romanian)

C.P. Niculescu, L.-E. Persson, Convex Functions and Their Applications, Springer-Verlag, 2006

C. Udrişte, Convex Functions and Optimization Methods on Riemannian Manifolds, Springer, Dordrecht, 1994

H. Bonnel, J. Collonge, Optimization over the Pareto outcome set associated with a convex bi-objective optimization problem: theoretical results, deterministic algorithm and application to the stochastic case, Journal of Global Optimization, 62(3) (2015), 481-505

H. Bonnel, L. Tadjihounde, C. Udrişte, Semivectorial Bilevel Optimization on Riemannian Manifolds, Journal of Optimization Theory and Applications, 167(2) (2015), 464-486

C. Drăguşin, Min-max pour des critères multiples, RAIRO Recherche Opérationnelle/Operations Research, 12, 2 (1978), 169-180

D. T. Norris, Optimal Solutions to the L_∞ Moment Problem with Lattice Bounds, PhD Thesis (2002),

University of Colorado, Mathematics Department, 2003

G. Cassier, Problèmes des moments sur un compact de R^n et décomposition des polynȏmes à plusieurs variables, Journal of Functional Analysis, 58 (1984), 254-266

O. Olteanu, Convexité et prolongement d’opérateurs linéaires, C. R. Acad. Sci. Paris, Série A, 286 (1978), 511-514

O. Olteanu, Sur les fonctions convexes définies sur les ensembles convexes bornés de R^n, C. R. Acad. Sci. Paris, Série A, 290 (1980), 837-838

O. Olteanu, Théorèmes de prolongement d’opérateurs linéaires, Rev. Roumaine Math. Pures Appl., 28, 10 (1983), 953-983

O. Olteanu, Application des théorèmes de prolongement d’opérateurs linéaires au problème des moments et à une généralisation d’un théorème de Mazur-Orlicz, C. R. Acad. Sci. Paris, Série I, 313(1991), 739-742

O. Olteanu, A strong separation theorem in normed linear spaces, Mathematica (Cluj), 35(58), 1 (1993), 59-63

O. Olteanu, New results on Markov moment problem, International Journal of Analysis, Vol. 2013, Article ID 901318, pp. 1-17. http://dx.doi.org/10.1155/2013/901318

O. Olteanu, J.M. Mihăilă, Extension and decomposition of linear operators dominated by continuous increasing sublinear operators, U.P.B. Sci. Bull. Series A, 80, 3 (2018), 133-144

Published
2019-08-31
How to Cite
Olteanu, O. (2019). Earlier and Recent Results on Convex Mappings and Convex Optimization. MathLAB Journal, 3, 136-148. Retrieved from http://purkh.com/index.php/mathlab/article/view/495
Section
Research Articles