Explore the theory, software, and impact of HiGHS (High Performance Software for Linear Optimization) in this 55-minute lecture by Julian Hall from the University of Edinburgh. Delve into optimization techniques, including primal and dual simplex algorithms, hyper-sparsity, and parallel solution methods for structured and general linear programming problems. Learn about the HiGHS team, solver capabilities, and practical applications in linear programming. Examine optimality conditions, computational aspects of the simplex method, and the effectiveness of various approaches. Discover the performance of HiGHS in simplex, interior point, and mixed-integer programming scenarios, gaining insights into state-of-the-art optimization software and its real-world impact.