Convex optimization Winter 2019/20

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Final exam part 1 question list

01) Define convex cone, proper cone, dual cone
02) Define convex set, show that an given set is convex from the definition
03) Operations that preserve convexity (list at least 3)
04) Definition of conical hull, convex hull
05) Support hyperplane theorem for convex sets -- what does it say, sketch the situation
06) Define convex function, show that a given function is convex using the definition
07) Properties of first and second derivatives smooth convex functions
08) Properties of the minimum of convex function
09) Operations on functions that preserve convexity
10) Define concave function, log-convex function, quasiconvex function
11) Define convex optimization problem
12) Define feasible and optimal solution
13) What is the feasibility problem? How does this problem relate to optimization problems?
14) Define LP, QP, SOCP
15) Define SDP
16) Define the cone of positive semidefinite matrices, give at least 1 application
17) Define a cone program (program with generalized inequalities as constraints)
18) Define a vector optimization problem, explain what a Pareto optimal solution is
19) Explain what scalarization is, why it is useful
20) Explain the main idea of how to approximate the optimal solution of a quasiconvex program by solving a series of convex programs
21) Define dual program (including Lagrangian and Lagrange dual function)
22) Weak duality theorem -- statement and applications
23) State the Slater criterion and strong duality theorem
24) Give at least 2 applications of strong duality theorem
25) Sensitivity of optimal value to initial conditions: State the 2 results we had and explain what they mean in practice
26) State the Farkas lemma, give 1 example of application
27) Explain what complementarity is, give an example of how to use it to compute the primal optimal solution if we know the dual optimal solution
28) State the KKT conditinos, explain why they are important
29) Norm minimization problem -- what it is, 1 example
30) Robust norm minimization -- what it is, 1 example
31) Support Vector Machine -- what it is, what is the “support”
32) Maximum likelihood method -- what it is, how to use it
33) Maximum a posteriori probability method -- what it is, how to use it
34) Löwner-John ellipsoid -- what it is, what is it for, how to calculate it
35) Define what a “descent method” means, how is it used for unconstrained minimization
36) Explain what a sparse matrix is, why are they useful
37) Explain what gradient descent is, what are its advantages and disadvantages
38) Explain what Newton's method is, what its advantages and disadvantages are
39) Give at least two ways of how to minimize a function subject to linear equality constraints
40) Give a sketch of interior point methods