مقایسهٔ روش‌ها

	بهینه‌سازی دو سطحی (رهبر-پیرو)	برنامه‌ریزی عدد صحیح (IP) و برنامه‌ریزی عدد صحیح مختلط (MIP)	برنامه‌ریزی غیرخطی	بهینه‌سازی استوار
حوزه	بهینه‌سازی	بهینه‌سازی	بهینه‌سازی	بهینه‌سازی
خانواده	Process / pipeline	Process / pipeline	Process / pipeline	Process / pipeline
سال پیدایش	1998	1958	2006	1970s theoretical roots; modern tractable form from late 1990s–2004
پدیدآور	Jonathan Bard	Ralph Gomory (cutting planes, 1958); land-and-doig branch-and-bound (1960)	Jorge Nocedal & Stephen Wright	Ben-Tal, El Ghaoui & Nemirovski (seminal book, 2009); Bertsimas & Sim (tractable polyhedral formulation, 2004)
نوع	Hierarchical mathematical programming	Mathematical optimisation — exact combinatorial method	Continuous mathematical optimization	Mathematical programming framework
منبع بنیادین	Bard, J. F. (1998). Practical Bilevel Optimization: Algorithms and Applications. Kluwer Academic Publishers. ISBN: 978-0-7923-5458-7	Wolsey, L.A. (1998). Integer Programming. Wiley. ISBN: 9780471283669	Nocedal, J., & Wright, S. J. (2006). Numerical Optimization (2nd ed.). Springer. ISBN: 978-0-387-30303-1	Ben-Tal, A., El Ghaoui, L. & Nemirovski, A. (2009). Robust Optimization. Princeton University Press. ISBN: 9780691143682
نام‌های دیگر	Stackelberg Optimization, Hierarchical Programming, Nested Optimization, İki Düzeyli Optimizasyon	IP, MIP, mixed-integer programming, mixed-integer linear programming	NLP optimization, Constrained nonlinear optimization, Smooth optimization, Doğrusal olmayan programlama	minimax optimization, worst-case optimization, Gürbüz Optimizasyon (Robust Optimization)
مرتبط	3	4	3	5
خلاصه	Bilevel optimization is a class of mathematical programming problems in which one optimization problem is nested inside another. The upper-level (leader) problem optimizes its objective subject to constraints that include the solution of a lower-level (follower) problem. Formalized comprehensively by Jonathan Bard in 1998, the framework models hierarchical decision-making where the leader anticipates and accounts for the rational response of the follower.	Integer programming (IP), also called mixed-integer programming (MIP) when only some variables are restricted to whole numbers, is a branch of mathematical optimisation in which some or all decision variables must take integer or binary values. Building on linear programming, it was formalised through Ralph Gomory's cutting-plane method (1958) and the Land-and-Doig branch-and-bound algorithm (1960), and it has since become the standard exact framework for scheduling, assignment, routing, and resource-allocation problems.	Nonlinear programming (NLP) is a branch of mathematical optimization concerned with problems in which the objective function or at least one constraint is nonlinear. Formalized comprehensively by Jorge Nocedal and Stephen Wright in their seminal 2006 text, NLP encompasses gradient-based algorithms — including sequential quadratic programming (SQP), interior-point methods, and quasi-Newton approaches — for finding locally or globally optimal solutions to continuous decision problems arising across engineering, economics, and the physical sciences.	Robust optimization is a mathematical programming framework, formalised by Ben-Tal and Nemirovski in the late 1990s and made broadly tractable by Bertsimas and Sim (2004), that finds decisions guaranteed to perform acceptably under every scenario within a predefined uncertainty set — rather than assuming parameter values are known exactly. Instead of optimising for a single expected outcome, it minimises the worst-case objective across all plausible realisations of uncertain data.