Sharif Digital Repository

We consider sequential quadratic programming (SQP) methods applied to optimization problems with nonlinear equality constraints and simple bounds, recently introduced in the literature. In particular, we describe and analyze a truncated SQP algorithm due to Izmailov and Solodov in which subproblems are solved approximately by an infeasible predictor-corrector interior-point method, followed by setting to zero some variables and some multipliers so that complementarity conditions for approximate solutions are satisfied. Verifiable truncation conditions based on the residual of optimality conditions of subproblems are explained to ensure both global and fast local convergence. Global... 

In this thesis we review several arithmetical questions about the diophantine equation φ[x]=q where φ is a nondegenerate symmetric bilinear form on a finite dimensional vector space, φ[x] is the corresponding quadratic form and q is a nonzero element of the ground field. We obtain class number and mass formulae for the orthogonal group associated to φ in a number field utilizing the concept of quadratic forms, lattices and adelic language  

In this paper Alexander Berkovich and Hamza Yesilyurt revisit old conjectures of Fermat and Euler regarding the representation of integers by binary quadratic form x2 + 5y2. Making use of Ramanujan’s 1 1 summation formula, they establish a new Lambert series identity for Σ1 n;m=1 qn2+5m2 . Conjectures of Fermat and Euler are shown to follow easily from this new formula. But they do not stop there. Employing various formulas found in Ramanujan’s notebooks and using a bit of ingenuity, they obtain a collection of new Lambert series for certain infinite products associated with quadratic forms such as x2+6y2, 2x2+3y2, x2+15y2, 3x2+5y2, x2+27y2, x2+5(y2+z2+w2), 5x2+y2+z2+w2. In the process, they...