Convex programming issues are fundamental to many fields of science and engineering, from signal processing and control systems to machine learning and data analysis. Due to its extensive applicability and influence, the effective resolution of these problems is of paramount importance. In this article, we give a thorough investigation into the creation and improvement of optimisation algorithms specifically suited for convex programming issues. We first explore the theoretical underpinnings of convex optimisation and talk about the characteristics that make these issues accessible to effective methods for solution. We emphasise the significance of convexity, duality, and optimality conditions that direct the creation of successful optimisation techniques. The design concepts and optimisation strategies for first-order methods, which depend on gradient information, are next examined. We discuss more complex techniques including accelerated gradient methods and proximal algorithms, as well as traditional algorithms like gradient descent and its variations. We go over their computational challenges, convergence characteristics, and accuracy vs. speed trade-offs. On benchmark convex programming problems, we give numerical tests and comparative analyses to assess how well the suggested optimisation strategies perform. We go over their advantages and disadvantages as well as the consequences of our research for practical use. This paper offers a thorough examination of the design and optimisation of algorithms specifically suited for resolving these issues, which advances the subject of convex programming as a whole. For practitioners and scholars working on convex optimisation, the offered methodologies provide insights and guidelines, supporting the effective resolution of challenging issues across a variety of disciplines.
Key words: Convex Programming, Proximal Algorithm, Gradient descend method, complex technique.
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