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Lec12P1, Orf363/Cos323

Di: Everly

Start by introducing a multiplier for each constraint: First, we need to preserve the inequalities after multiplication. This problem is called the dual LP! with optimal value 1900. Bingo! They

Lecture 12: Duality + robust linear programming. Lecture 13: Semidefinite programming + SDP relaxations for nonconvex optimization. Lecture 14: A working knowledge of computational

The Idea Behind Duality Lec12p1, ORF363/COS323 Lec12p1, ORF363/COS323 This lecture Instructor: Amir Ali Ahmadi Fall 2014 Linear programming duality + robust linear programming

New: all lecture notes and problems sets of Fall 2015 in one file: [pdf] (currently taken down) The lecture notes below summarize most of what I cover on the blackboard during class. Please

The goal of this lecture is to refresh your memory on some topics in linear algebra and multivariable calculus that will be relevant to this course. You can use this as a reference

In the last couple of lectures we have seen several types of gradient descent methods as well as the Newton’s method. Today we see yet another class of descent methods that are particularly

In 1947, Dantzig invented the first practical algorithm for solving LPs: the simplex method.

In the previous lecture, we saw the general framework of descent algorithms, with several choices for the step size and the descent direction. We also discussed convergence issues associated

Recall that this would suffice for global optimality if is convex. We now begin to see some algorithms for this purpose, starting with gradient descent algorithms. These will be iterative

Lec9p1, ORF363/COS323 Instructor: Amir Ali Ahmadi Fall 2014 This lecture: • Multivariate Newton’s method TAs: Y. Chen, G. Hall, • Rates of convergence J.

SDP is a very natural generalization of LP. It is still a convex optimization problem (in the geometric sense). We can solve SDPs efficiently (in polynomial time to arbitrary accuracy).

One-dimensional line search will be used as a subroutine in future lectures for multivariate optimization. Some algorithms that we see here (e.g., Newton’s method) will directly generalize

Recall that this would suffice for global optimality if is convex. We now begin to see some algorithms for this purpose, starting with gradient descent algorithms. These will be iterative

Today we start off by proving results that explain why we give special attention to convex optimization problems. In a convex problem, every local minimum is automatically a global

• In the previous lecture, we covered some of the reasons why convex optimization problems are so desirable in the field of optimization. We also gave some characterizations of convex

Lec11p1, ORF363/COS323 Lec11 Page 1 . Applications of linear programming Example 1: Transportation All plants produce product A (in different quantities) and all warehouses need

Lec10p13, ORF363/COS323 Lec10 Page 13 . A bit of Leontief history Source: [Lay03] In 1949, Wassily Leontief (then at Harvard) used statistics from the U.S. Bureau of Labor to divide the

Lec12p1, ORF363/COS323 Lec12 Page 1 . Let’s systematize what we did in this example. Start by introducing a multiplier for each constraint: •First, we need to preserve the inequalities after

Lec12p1, ORF363/COS323 Lec12 Page 1 . Let’s systematize what we did in this example. Start by introducing a multiplier for each constraint: •First, we need to preserve the inequalities after

Lec2p1, ORF363/COS323 Lec2 Page 1 . Inner products and norms Definition of an inner product •Positivity: and iif x=0. •Symmetry: •Additivity: •Homogeneity : An inner product is a real-valued

Lec12p1, ORF363/COS323 Lec12 Page 1 . Let’s systematize what we did in this example. Start by introducing a multiplier for each constraint: •First, we need to preserve the inequalities after