TAILIEUCHUNG - On invex programming problem in hilbert spaces
In this paper we introduce the invex programming problem in Hilbert space. The requisite theory has been established to characterize the solution of such class of problems. | Yugoslav Journal of Operations Research 25 (2015), Number 3, 379–385 DOI: ON INVEX PROGRAMMING PROBLEM IN HILBERT SPACES Sandip CHATTERJEE Department of Mathematics,Heritage Institute of Technology, Kolkata-700107, West Bengal,India functionals@ Rathindranath MUKHERJEE Department of Mathematics, The University of Burdwan, Bardhaman-713104, West Bengal,India rnm bu math@ Received: October 2014 / Accepted: March 2015 Abstract: In this paper we introduce the invex programming problem in Hilbert space. The requisite theory has been established to characterize the solution of such class of problems. Keywords: Invexity, Compactness, Weak topology, Frechet derivative. MSC: 26B25, 26A51, 49J50, 49J52. 1. INTRODUCTION The mathematics of Convex Optimization was discussed by several authors for about a century [2, 3, 4, 5, 9, 10, 15, 17, 23, 24]. In the second half of the last century, various generalizations of convex functions have been introduced [2, 3, 4, 5, 6, 7, 10, 11, 12, 14, 16, 18, 19, 20, 22]. The invex(invariant convex), pseudoinvex and quasiinvex functions were introduced by in 1981 [14]. These functions are extremely significant in optimization theory mainly due to the properties regarding their global optima. For example, a differentiable function is invex iff every stationary point is a global minima[6]. Later in 1986, Craven defined the non-smooth invex functions [11]. For the last few decades generalized monotonicity, duality and optimality conditions in invex optimization theory have been discussed by several authors but mainly in Rn [6, 11, 12, 14, 18, 19, 20]. The basic 380 , / Invex Programming Problem difficulty of genaralizing the theory in infinite dimensional spaces is that, unlike the case in finite dimension, closedness and boundedness of a set does not imply the compactness. However, in reflexive Banach spaces the problem can be alleviated by working with weak .
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