Practice problem set: Please find here a growing list of practice problems, including some previous year's question papers and their solutions. In particular, do look at this set. You can also refer to this wiki for definitions of several concepts relevant to this course, connections between those concepts through theorems and proofs of those theorems.

It is very important that you attempt the homework problems that were posted at the end of (almost) each lecture.

You can consider solving the following problems from Boyd (solutions to few of them has been discussed in class or moodle): (A) 2.1 to 2.37 and 2.39 (B) 3.1 to 3.35 (question involving quasi-convexity are optional) and 3.47 to 3.56. (C) 4.1 to 4.6 and 4.8 to 4.20 and 4.8 to 4.29 and 4.38 to 4.49 and 4.56, 4.61, 4.63. (D) 5.1, 5.2, 5.4 (discussed in midsem), 5.5, 5.7, 5.9, 5.10 to 5.13 (partly discussed in midsem), 5.14 to 5.20, 5.21 to 5.23, 5.26 to 5.31 and 5.39, 5.41 to 5.43 (E) 9.1 to 9.11 and 9.23 to 9.32. 9.12 to 9.22 are optional (F) 11.1 to 11.12, 11.15 to 11.18

• classnotes
• You can refer to Linear Algebra Done Right for brushing up/reading up on linear algebra.
• Given an affine set S , give a procedure to obtain A and b such that S = { x | Ax = b } following the hints given at page 5 of the classnote.
• Please notice the definition of an ellipsoid in the note. Is P being positive definite is necessary for convexity of ellipsoid?
• List down all properties of positive definite matrices that you think would be useful in this course

• classnotes
• Refer to Chapter 2 (especially pages 93 to 97) of Topology by Munkres to understand the connection between open sets, closure, closed sets, limit points and neighborhoods for topological spaces. Topology and Groupoids by Ronald Brown is another good reference.
• Prove that the real part of the eigenvalue of a (not necessarily symmetric) positive definite matrix is always positive.
• Understand the equivalence between the different definitions for interior, boundary, open and closed sets in the case of normed spaces.
• Read how an inner product space is a normed space.

• classnotes
• Show that the following are vector spaces (assuming scalars come from a set S), and then answer questions that follow for each of them: Set of all matrices on S, set of all polynomials on S, set of all sequences of elements of S. (HINT: You can refer to this book for answers to most questions in this homework.) How would you understand the concepts of independence, span, basis, dimension and null space (chapter 2 of this book), eigenvalues and eigenvectors (chapter 5), inner product and orthogonality (chapter 6)? EXTRA: Now how about set of all random variables and set of all functions. Deadline: January 23 2015.

• classnotes
• For the vector spaces (along with their specific norms/inner products) discussed in class, identify some linear maps and corresponding null spaces. (HINT: You can refer to chapter 3 of this book). Deadline: January 27 2015.
• We discussed different kinds of general vector spaces and different norms on each vector space. Present 5 SPECIFIC examples of convex norm balls or norm cones based on this discussion from vector spaces other than R^n. Can you interpret them? Deadline: January 27, 2015
• (Pending from previous class) Prove that sqrt(x'Px) is a norm for vector x, if P is positive definite. Thus, an ellipsoid is a norm ball. Deadline: January 27 2015.
• Extra: Read up about Strictly convex space and how strict convexity is somewhere between inner product space and a general normed space.

• classnotes
• Refer to chapter 2 of the book "Duality in Vector Optimization" entitled "Preliminaries on convex analysis and vector optimization") and read up the concepts of affine (linear) map, algebraic dual spaces, algebraic dual cones, topological dual spaces and topological dual cones Deadline: January 30 2015.

• classnotes
• References: "Preliminaries on convex analysis and vector optimization" pages 1-17, Boyd and Vandenberghe Section 2.6.1, Nemirovski: Lecture Notes on Modern Convex Optimization Sections B.1.4, B.2.6.B,
• What are the (topological) conic duals of a positive semi-definite cone and of a positive definite cone? What is their boundary? Interpret them. Deadline: February 3, 2015.
• Prove properties 4 and 6 on page number 9 of the notes. Deadline: February 3, 2015.

• classnotes
• References: Boyd and Vandenberghe Sections 2.2.5, 2.4, and 2.6.1, Nemirovski: Lecture Notes on Modern Convex Optimization Sections B.2.5.C.1, B.2.7.B.
• References connecting today's class to next class: "Advances in Convex Optimization: Conic Programming" Sections 1 and 2. The trajectory is roughly as follows: Linear program (LP) --> weak duality --> dual LP --> Generalised inequality --> Proper cones --> Conic Program (CP) --> Weak duality for CP --> Dual for CP using dual cone (Semi-definite program and LP are special cases of Conic Programs)
• Write code to plot the "dual" representation of a (2 by 2) positive definite cone described on page 3 of the notes at notes/enotes/2015-7.pdf. You may approximate the representation (as discussed in class) based on (a) polar coordinate representation and (b) choosing a finite number of half-spaces. Deadline: February 8, 2015 (to be submitted here).
• With reference to pages 7 and 8 of the notes, try to prove (even partially) that K being a proper convex cone is a necessary and sufficient condition for the corresponding induced inequality to be a well-defined partial order. Deadline: February 6, 2015.

• classnotes
• References: Boyd and Vandenberghe Sections 2.4, and 2.5, 2.6, Nemirovski: Lecture Notes on Modern Convex Optimization Sections 1.4.1, 1.4.2, 1.4.3.
• What is the dual of the norm cone? (See page 15 of the notes for a start). What specifically is the dual of the norm cone (or the dual norm) when the norm is the p-norm in a Eucledian space? Deadline: February 10, 2015.

• classnotes
• References: Boyd and Vandenberghe Sections 2.6, 5.2 (especially page 225), 5.8 (especially page 263), Nemirovski: Lecture Notes on Modern Convex Optimization Sections 1.4.5 (esp theorem 1.4.2), 1.4.4 (extra reading for geometric interpretation of conic dual).
• Refer to pages 11 and 12 of the class notes. Fill in the green spaces. Deadline: February 13, 2015.

• classnotes
• References: Boyd and Vandenberghe Sections 2.6, 5.2 (especially page 225), 5.8 (especially page 263), Nemirovski: Lecture Notes on Modern Convex Optimization Sections 1.4.5 (esp theorem 1.4.2), 1.4.4 (extra reading for geometric interpretation of conic dual).
• Extra reading for next class: Geometric interpretation of duality in optimisation problems. Section 4.4.3 (page 292 onwards) of my lecture notes and specifically for conic duality, Nemirovski Section 1.4.4.
• Refer to pages 12 and 13 of the class notes for proof of strong conic duality theorem. Fill in the green spaces. Deadline: February 17, 2015.

• classnotes
• References: Boyd and Vandenberghe Sections 5.1 (before Section 5.1.5), 5.2 (before Section 5.2.3), 5.8 (especially page 263), Nemirovski: Lecture Notes on Modern Convex Optimization Sections 1.4.5 (esp theorem 1.4.2)
• OPTIONAL READING: Geometric interpretation of duality in optimisation problems: Boyd and Vandenberghe Section 5.3, Page 10 to 13 of class notes, Section 4.4.3 (page 292 onwards) of my lecture notes and specifically for conic duality, Nemirovski Section 1.4.4.
• (Pending from previous class) Prove that S = {(x1,x2) | x1x2 >= 1, x1>0, x2>0} is a closed convex set, where x1 and x2 are real valued. Represent S as an intersection of half-spaces. Deadline: February 27, 2015

• midsem solutions
• classnotes
• Geometric interpretation of duality in optimisation problems: Boyd and Vandenberghe Section 5.3, Page 10 to 13 of class notes, Section 4.4.3 (page 292 onwards) of my lecture notes and specifically for conic duality, Nemirovski Section 1.4.4.
• Suppose the set I discussed in the class is convex. Can you derive necessary and/or sufficient conditions that the functions f and gi's should satisfy? What about hj's? Deadline: March 3, 2015

• classnotes
• Geometric interpretation of duality in optimisation problems: Boyd and Vandenberghe Section 5.3, Page 10 to 13 of class notes, Section 4.4.3 (page 292 onwards) of my lecture notes and specifically for conic duality, Nemirovski Section 1.4.4.
• Complete the proof of equivalence between (a) and (b) on page 3 of the notes Deadline: March 10, 2015
• On page 7 of the notes, we invoked a result that a Convex set has an empty interior if and only if there exists a hyperplane containing the set. Try and prove this result Deadline: March 10, 2015

• classnotes
• Boyd and Vandenberghe Section 2.3, 2.5.1, 2.5.2, 3.1.1, 3.1.7, 3.6.2, Nemirovski Section B.2.1.
• Roughly, submodularity is for discrete optimisation what convexity is for continuous optimisation. Check this out
• Prove that a vector valued function is K-convex if and only if its epigraph (w.r.t generalized K-inequality) is a convex set. See page 38 of the notes. Deadline: March 17, 2015
• Try and reason out the condition under which the epigraph of a function will be closed? See page 38 of the notes. Deadline: March 17, 2015

• classnotes
• Boyd and Vandenberghe Section 3.1.1, 3.1.2, 3.1.3, 3.1.4, 3.1.5, 3.1.6, 3.1.7, Nemirovski Section C.6.1.
• Construct a convex function which has discontinuities on the relative boundary of its domain (refer to page 14 of class notes). Deadline: March 20, 2015
• We saw that a function f from R^n to R is convex if and only if the restriction of f to any line is convex. Does the same hold for closedness? Prove. Deadline: March 20, 2015
• Read up pages 231 to 239 of the notes. If you feel the need, also brush up optimization results for functions of single variables on page 216 to 230 from these notes. Deadline: March 20, 2015

• classnotes
• Sections 4.1.4, 4.1.5 and 4.2.9 of our notes. Boyd and Vandenberghe Section 3.1.1, 3.1.6, 3.1.7, 3.1.8, 3.2, Nemirovski Section C.6.1

• classnotes
• Sections 4.1.4, 4.1.5 and 4.2.9 of our notes. Boyd and Vandenberghe Section 3.1.3, 3.1.6, 3.1.7, Nemirovski Section C.6.2.
• Explain the procedure for computing the subgradient of the function f(x):R^n->R given by f(x) = ||x||_1 (the 1-norm of x) Deadline: March 24, 2015

• classnotes
• Sections 4.2.9 and 4.2.10 of our notes. Boyd and Vandenberghe Section 3.1.3, 3.1.4, 3.1.5, Nemirovski Section C.6.2.
• Try to derive a closed form expression for the subdifferential of the 1-norm function (see page 11 of class notes) Optional
• Complete proof on page 12 of the notes that the subdifferential is nonempty in the relative interior of the domain of a convex function Deadline: March 27, 2015

• classnotes
• Boyd and Vandenberghe Section 3.1.3, 3.1.4, 3.1.5, Nemirovski Section C.6.2.

• classnotes
• classnotes
• Boyd and Vandenberghe Sections 9.1, 9.2, 9.3, 9.4, 9.4.1, Nemirovski Section C.6.2.

• classnotes
• classnotes
• Boyd and Vandenberghe Sections 4.2.4, Nemirovski Sections 5.3.1 and 5.3.2 (a more genral reading). Section 4.4.1 of our notes.
• Complete derivation of the proximal gradient steps for the problem on page 16 of these notes Deadline: April 7, 2015
• Prove that the if a convex function is differentiable, the only subgradient (in the interior) is the gradient (refer to page 16 of notes) Deadline: April 7, 2015
• Prove the theorem 76 on page 1 of the class notes. Deadline: April 7, 2015

• classnotes
• Please find proof of theorem 76 in the class notes.
• Boyd and Vandenberghe Sections 5.1, 5.2, 3.3, 5.1.6, 5.7.1. Section 4.4.1 of our notes. Also see these notes. A more general reading: Section 5.3.2 of Nemirovski. A more advanced reading can be found here

• classnotes
• Boyd and Vandenberghe Sections 5.5. Sections 2.1, 2.2, 2.3 and 3.1 of this paper. Section C.6.1 of Nemirovski. Extra reading: Chapters 1, 2,3 and 4 of these notes. For extra reading, also see the various links on the last two pages of the class notes.

• classnotes
• Boyd and Vandenberghe Sections 11.1-11.4, Section 16.5 of Nocedal and Wright, Extra reading: Sections 4.2 and 4.3 of Nemirovski. Also see these notes and their explanations given in the class.

• classnotes
• Nemirovski Section C.4 (appendix)
• classnotes
• Boyd and Vandenberghe Section 9.3.1. Extra advanced reading: Nemirovski Section 5.4.2
• classnotes
• Nocedal and Wright Section A.2 (subsection "Rates of Convergence").

• classnotes
• Nemirovski Section C.4 (appendix)