2. Nonlinear programming:
Examples for optimization problems. Classification of optimization problems. Numerical solution of one variable problems. Secant method, bisection method. Unconstrained problems. Types of the extremum. Optimality conditions. First- and second order necessary conditions, second order sufficient condition. Weierstrass theorem. Level sets, coercivity. Convexity. Convex functions and their properties. Local-global minimum property. Characterization of differentiable convex functions. Quadratic functions.Least square problems (linear and nonlinear). Gradient method. Step size rules. Newton's method, damping, rate of the convergence. Constrained problems. Optimality conditions. Lagrange multipliers. KKT conditions. Tangent cone, linearized tangent cone. Constraint qualifications. Trust region methods. Duality theory. Strong duality, weak duality. Convex case. Geometric programming. Penalty function, quadratic penalty. Quadratic programming.
Suggested books:
• J. Nocedal and S. Wright: Numerical optimization, Springer, 2006.
• M. Ulbrich and S. Ulbrich: Nichtlineare optimierung, Birkhäuser, 2012.
• Amir Beck: Introduction to nonlinear optimization: theory, algorithms, and applications with Matlab. MOS-SIAM Series on Optimization, 2014.
• A. Mircea: Practical optimiztaion with Matlab, Cambridge Scholars Publishing, 2019.

3. Stochastic programming:
Decision under uncertainty, Bernoulli decision. Probability bounds of union of events. Markov decision process. Bellmann equations; Value and policy iteration algorithms. Theory of the logconcave measures. Theory and numerical solution techniques of programming problems under individual or joint probabilistic constraints. Models with random objective function. Dynamic stochastic programming models. Two-stage stochastic programming problems and their mathematical properties. Solution of the two-stage stochastic programming problems by basis decomposition technique when the random vector of the problem has discrete distribution; Multi-stage stochastic programming problems; Stochastic gradient methods; Gauss process, Bayes optimization. Selected applications of the stochastic programming. News vendor decision problem, optimal control of a storage level; Queuing problems; Electric power generation capacity expansion under uncertainty. System reliability computation. Example for optimal control of reservoirs. PERT optimization problem. Optimization problems in finance.
Suggested books:
• Kall, Peter, Stein W. Wallace, and Peter Kall. Stochastic programming. Vol. 5. Chichester: Wiley, 1994.
• Kail, Peter, and Janos Mayer. "Stochastic linear programming, models, theory, and computation." (2005).
• Birge, John R., and Francois Louveaux. Introduction to stochastic programming. Springer Science Business Media, 2011.

4. Integer programming:
Basic models: knapsack problem, set covering and decomposition problem, quadratic assignment problem, traveling agent problem. Industrial Applications I: Deployment problems. Placement of machines and cells. Placement of integrated circuit elements in fixed positions. Distinguishing deployment problems from similar but different nature problems. Industrial Applications II: Exact models of scheduling problems. Industrial Applications III: Application of the traveling agent problem in various industrial and commercial environments. Cut-type methods I: The Gomory method. Cut-type methods II: Faces of the polyhedron of the traveling agent problem. Cut-type methods III: Intersection cut Counting methods Constraint and separation. Separation and cutting; cutting and separation; separation and pricing. Dynamic programming, Bellman principle. Group theory method. Metaheuristics. Duality. Lagrange multipliers. Benders decomposition.
Suggested books:
• A. Schrijver, Theory of linear and integer programming, John Wiley and sons, 1986

5. Combinatorial optimization:
Network flows, Ford-Fulkerson algorithm. Characterization of even graphs, maximum size matching in even graphs. Kőnig, Frobenius and Hall theorems. Graph colorings. Lower and upper bounds on the chromatic number. Egerváry's algorithm. Fourier-Motzkin elimination. Approximation algorithms. Scheduling problems.
Suggested books:
• B. Korte, J. Vygen, Combinatorial optimization, Springer, 2018.

6. Semi-definite optimization:
General convex optimization. Cone optimization. SDP duality theory. SDP and interior point algorithm. Sedumi, Yalmip, solvers on Neos server. Polynomial optimization, SOS.
Suggested books:
• E. de Klerk: Aspects of Semidefinite Programming, Kluwer Academic Publisher, 2002.
• H. Wolkowicz, R. Saigal, L. Vandenberghe: Handbook of Semidefinite Programming - Theory, Algorithms, and Applications, Springer, 2000.

7. Global optimization:
Needed tools from functional analysis. Banach spaces, dual spaces, weak* topology. Hahn-Banach theorem. Modern theory of convex functions. Lower semicontinuous functions. Fenchel conjugate, biconjugate, subdifferential, approximate subdifferential. Characterization of the global optimum. Ekeland variational principle and its applications. DC optimization. Toland-Singer duality theorem. Quadratic optimization. Conjugate gradient method. Quasi-Newton methods. Multiobjective optimization. Metaheuristic algorithms. Simulated annealing, tabu search, variable neighborhood search, genetic algorithm, particle swarm optimization, ant optimization.
Suggested books:
• J. Nocedal and S. Wright: Numerical optimization, Springer, 2006.
• R. Horst and P. M. Pardalos: Handbook of global optimization, Springer, 1995.
• Y. Nesterov: Lectures on convex optimization, Springer, 2018.
• P. Cortez: Modern optimization with R, Springer, 2014.
• I. Ekeland and R. Temam: Convex analysis and variational problems, Elsevier, 1976.
• Sean Luke: Essentials of Metaheuristics, Creative Commons, San Francisco, 2016.

8. Convex analysis:
Algebraic and topological properties of convex sets. Duality relations: separation theorems, polar, dual operators. Convex/linear inequality systems, extremal points and sets. Differentiation of convex functions: directional derivative, subgradient, continuity and monotonicity. Duality theory of convex optimization problems. Fenchel's duality. Maximum of convex functions. Saddle functions: continuity, differentiability. Theory of minimax problems. Convex algebra.
Suggested books:
• R.T. Rockafellar, Convex Analysis, Princeton University Press, 1970
• J.M. Bornwein, A.S. Lewis, Convex Analysis and Nonlinear Optimization, Springer, 2006
• I. Ekeland, R. Temam, Convex analysis and variational problems, Elsevier, 1976

9. Dynamic programming:
Optimal strategies, discrete models. The basic principle of dynamic programming. Unfavorable and favorable games, bold and cautious strategies. Optimal parking, large-scale procurement planning. Lagrange mechanics, Hamilton-Jacobi equation. Viscous approximation, Hopf-Cole transformation, Hopf-Lax's infimum-convolution formula. Deterministic optimal control, optimal investment strategy, viscous solutions of the general Hamilton-Jacobi equation. Pontryagin's maximum principle, conditional edge search in function space. Optimal control of stochastic models, the Hamilton-Jacobi-Bellman equation.
Suggested books:
• L.C. Evans: Partial Differential Equations, AMS, Providence, R.I., 1998.

10. Econometrics:
Introduction to econometrics. Bivariate relationships: linear regression, least squares (LS) estimation and its statistical properties, Gauss-Markov theorem, prediction. Multivariate linear regression for uncorrelated, identically standard errors and general error processes, general Gauss-Markov theorem, forecasting, multi-collinearity. Generalized LS method, special cases (autocorrelated noise, uncorrelated noise with different standard deviations), auxiliary variables (IV) method. Time series analysis: stationarity, autocorrelation, white noise process, special models (linear filters, autoregressive (AR) process, moving average (MA) process, ARMA processes). Parameter estimation (ML-estimation), forecasting. Integrated and cointegrated processes (ARIMA models), trend, seasonality. Spectral representation, periodogram and its estimation, spectrum estimation. Multivariate models: VAR(1) processes, n-dimensional ARMA processes, stationarity, stability, Lyapunov equation. Fractionally integrated processes, ARFIMA models, long-memory processes and their estimation. Stochastic volatility models: ARCH and GARCH processes, bilinear processes and their characteristics, stationarity, parameter estimation, state space representation. Applications: time series analysis of financial market returns, analysis of biological data.
Suggested books:
• J.H. Stock, M.W. Watson, Introduction to Econometrics, Pearson 2022.

11. Game theory:
Non-cooperative game theory: Non-cooperative games, normal form, mixed extension of finite games (introduction of mixed strategy), Nash equilibrium, proving the existence of (mixed) Nash equilibrium, the concept of correlated equilibrium and its relation to Nash equilibrium, the concept of rationalizability and its relation to Nash equilibrium, the method of iterative elimination of (strictly) dominated strategies and its relation to rationalizability, Games in tree form, rewriting tree form into normal form, reduced form, mixed and behavioral strategies, the subgame-perfect Nash equilibrium, the method of backward induction, Incomplete information games, imperfect information games, Bayesian games, ex-ante Bayesian–Nash equilibrium and interim Bayesian–Nash equilibrium and their relation, the relation between rationalizability and backward induction, Knowledge and public knowledge, private information (agreeing to disagree), mixed strategy interpretations, asymmetric information, sequential equilibrium.
Cooperative game theory: Cooperative games, types of cooperative games, Nash program, cooperative games with transferable utility, the concept of solution, the core, the necessary and sufficient condition for the core to be nonempty: the Bondareva–Shapley theorem, The concept of Shapley value, axiomatic approach, axiomatizations of the Shapley value, Prenucleus, nucleolus, axiomatization results for prenucleus and nucleolus, Additional solution concepts for cooperative games with transferable utility: stable set and bargaining sets, Cooperative games with nontransferable utility, the core, the sufficient condition for the core to be nonempty: the Scarf theorem, the necessary and sufficient condition for the core to be empty: Pi-equilibrium.
Suggested books:
• A. Rubinstein, J.O. Martin: A Course in Game Theory, MIT Press (1994)
• B. Peleg, P. Sudhölter: Introduction to the Theory of Cooperative Games, Springer (2007)

12. Control theory:
Introduction to control theory. Controllability of finite dimensional linear systems. Kálmán rank condition. Duality between controllability and observability. Local controllability of finite dimensional nonlinear systems. The linear test. Lie algebra condition. Application to driftless control affine systems. Global controllability of finite dimensional nonlinear systems. Lipschitz perturbations of controllable linear control systems. Global controllability and homogeneity. Stabilization of finite dimensional linear control systems. Pole-shifting theorem and applications. Some examples of stabilization of finite dimensional nonlinear systems with feedback controls.
Suggested books:
• L. C. Evans, An Introduction to Mathematical Optimal Control Theory, University of California, Berkeley 2010.
• D. Liberzon, Calculus of Variations and Optimal Control Theory: A Concise Introduction, Princeton Univ. Press, 2012.

13. Linear complementary problems:
Matrix analysis, pivotal algebra, matrix factorization, iterative methods for equations, linear inequalities, quadratic programming, pivoting methods, iterative methods.
Suggested books:
• R.W. Cottle, J.-S. Pang, R.E. Stone, The linear complementary problem, SIAM, 2009.

14.  Computational methods of Operations Research:
Operations Research models can be applied to a wide variety of problems in the real world. The goal of this course is to show the most well-known of these problems, and their LP / IP / MILP formulations. The main focus of the course will be the programming language called XPress Mosel, which is able to efficiently encode and solve Operations Research problems.
Topics planned: Oil refinery scheduling, Differences between LP/IP/MILP, The Knapsack problem, Linearization, Piecewise Linear approximations, Modelling Geometry problems, Minimax objective functions, Modelling with graphs, The Cutting Stock problem, Portfolio Selection, Quadratic Programming, Automatic Sudoku solving, Constraint Programming, Scheduling Problems, The Travelling Salesman problem, Nonlinear Programming, The Pooling problem.
Suggested books:
• Ruhul Amin Sarker, Charles S. Newton - Optimization modelling: A practical approach
• H. Paul Williams - Model Building in Mathematical Programming
• Leo Liberti - Problems and exercises in Operations Research