Mixed-integer nonlinear programs are typically solved using branch-and-bound algorithms. A key determinant of the success of such methods is their ability to construct tight and tractable relaxations. The predominant relaxation strategy used by most state-of-the-art solvers is the factorable programming technique. This technique recursively traverses the expression tree for each nonlinear function and relaxes each operator over a bounding box that covers the ranges for all the operands. While it is versatile, and allows finer control over the number of introduced variables, the factorable programming technique often leads to weak relaxations because it ignores operand structure while constructing the relaxation for the operator.<div>In this thesis, we introduce new relaxations, called composite relaxations, for composite functions by convexifying the outer-function over a polytope, which models an ordering structure of outer-approximators of inner functions. We devise a fast combinatorial algorithm to separate the hypograph of concave-extendable supermodular outer-functions over the polytope, although the separation problem is NP-Hard in general. As a consequence, we obtain large classes of inequalities that tighten prevalent factorable programming relaxations. The limiting composite relaxation obtained with infinitely many outer-approximators for each inner-function is shown to be related to the solution of an optimal transport problem. Moreover, composite relaxations can be seamlessly embedded into a discretization scheme to relax nonlinear programs with mixed-integer linear programs. Combined with linearization, composite relaxations provide a framework for deriving cutting planes used in relaxation hierarchies and more.<br></div>

The main purpose of the study was to understand preservice secondary mathematics teachers&rsquo / pedagogical content knowledge of composite and inverse functions. The study was conducted with three preservice secondary mathematics teachers in Graduate School of Education at Bilkent University. The instruments of the study were qualitative in nature and in four different types of data forms: observations, interviews, documents, and audiovisual materials. Observation data came from fieldnotes by conducting an observation of lessons participants taught at Private Bilkent High School. Interview data came from the transcriptions of semi-structured interviews. Document data came from survey of function knowledge, journal writings, vignettes, and lesson plans. Audiovisual data came from the examination of the videotape of the lessons participants taught. The findings reveal that preservice secondary mathematics teachers&rsquo / knowledge levels in components of pedagogical content knowledge were not at the desired levels and also they experienced difficulty while integrating that knowledge. The results of the study indicate that teacher education should provide courses that cover the content relevant to students in order to assure both depth and breadth in subject matter knowledge of the preservice teachers. Moreover, the activities which mimics the classroom cases and assures the integration of knowledge components like vignettes would be used in teacher education programs. Results can inform educational practices, and reforms in Turkey, and provide a basis for further research, with increased pedagogical content knowledge as the ultimate goal.