Over the past 40 years, the area of digital signal processing and filtering has undergone a rapid development. Today, most of the filtering tasks, that before were performed by using analog technology, have been substituted with inexpensive and, often, more reliable and flexible digital solutions. The optimization methods used today for digital filter design date back more than 50 years. Digital filters can mainly be divided into two main categories, Finite Impulse Response (FIR) filters and Infinite Impulse Response (IIR) filters. This thesis deals with the design of three different FIR structures; window functions for frequency analysis, antenna arrays, and a channel equalizer for mobile communication. For all three structures, the corresponding complex optimization problem can become semi-infinite, i.e. a finite number of unknowns with an infinite number of constraints. The Dual Nested Complex Approximation (DNCA) algorithm has been preferred to solve such complex optimization problems. This is partly motivated by the low computational cost and low memory consumption, which allows execution on any desktop or laptop computer. The thesis consists of three parts; Part I considers the design and enhancement of flattop windows constructed with a summation of shifted Dirichlet kernels; Part II deals with the design of antenna arrays, where the antenna element weights are complex-valued; and Part III, finally, employs semi-infinite quadratic programming to design a channel equalizer with a complex-valued response, i.e. with a non-linear phase.