A list of questions on Boyd & Vandenberghe

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_{Work in progress. To be updated whenever the opportunity arises.}

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I take the liberty of compiling a list of questions on Math SE pertaining to theorems, lemmas, examples and exercises from Boyd & Vandenberghe's Convex Optimization. Please consider linking to this list whenever someone posts a new question related to Boyd & Vandenberghe, for it may be a duplicate.

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Chapter 2

• Boyd & Vandenberghe, page 27 — intuition and proof for how a ray is convex rather than affine

• https://math.stackexchange.com/q/3918952/339790

• Boyd & Vandenberghe, problem 2.10 — is $A \in \mathbb S^n$ redundant?

• Boyd & Vandenberghe, problem 2.10 — is my solution correct?

• Boyd & Vandenberghe, example 2.2 — interior and relative interior points

• Boyd & Vandenberghe, example 2.13 — what is the "convex set of joint probabilities" in this example?

• Boyd & Vandenberghe, exercise 2.15 — on $\mbox{quart}$

• Boyd & Vandenberghe, Exercise 2.15g — Convexity of a region on the probability simplex

• Boyd & Vandenberghe, exercise 2.10 — two questions

• Boyd & Vandenberghe, exercise 2.12 — How to prove this set is convex?

• Boyd & Vandenberghe, question 2.31(d) — stuck on simple problem regarding interior of a dual cone

• Boyd & Vandenberghe, example 2.12 — image of the unit Euclidean ball under affine mapping

• Separability of disjoint convex sets through hyperplane

• How to prove the following for inner product and positive semidefinite matrices?

• Finding dual cone for a set of copositive matrices

• Boyd & Vandenberghe, problem 2.10 — sublevel set of quadratic is convex

• A confusion related to the rank of the matrix

• Boyd & Vandenberghe, problem 2.27 — converse supporting hyperplane theorem

• Boyd & Vandenberghe — Describing simplex as a polyhedron

• Proof of supporting hyperplane theorem in Boyd and Vandenberghe

• Show that if a convex $C$ has a supporting hyperplane at every point of its boundary, then it's convex

• How to prove the following for inner product and positive semidefinite matrices?

• Why does the set of positive definite matrices define a half-space?

• Definition of ellipsoid

• Set of symmetric positive semidefinite matrices is closed

• How to prove $B-A \succeq 0 \Leftrightarrow$ ellipsoid $x^TBx \leq 1$ contains $x^TAx \leq 1$?

• Disjoint convex sets that are not strictly separated

• Solution set of an LMI is convex

• Intuition for why a convex set with empty interior lies in an affine set

• How to prove the convexity of this set by the definition of convex set？

• Prove or disprove that the set of probabilities $\{p \in \Delta^n \mid \mbox{Var}(X) \leq \alpha\}$ is a convex set.

• How is a line passing through the origin a convex cone?

• Proving that the probabilities for which the variance of a distribution is greater than some number is convex

• Theorem of alternatives for strict generalized inequalities

Chapter 3

• https://math.stackexchange.com/q/4649061/339790

• Boyd & Vandenberghe, exercise 3.43 — First order condition for quasiconvex functions

• Boyd & Vandenberghe, example 3.34 — quasiconvex example

• Boyd & Vandenberghe, example 3.4 — question on Schur complements and LMIs

• Boyd & Vandenberghe, Example 3.10 — Convexity of $X \mapsto \lambda_ {\max}(X)$

• Boyd & Vandenberghe, example 3.33 — How to prove the Euclidean ball is convex?

• Boyd & Vandenberghe, problem 3.40 — on the implicit function theorem

• Boyd & Vandenberghe, problem 3.50 — which point am I missing?

• Boyd & Vandenberghe, problem 3.51 — how to show that the number of roots is even?

• How to show that product over sum function is log concave?

• A function is quasilinear if its domain and all level sets are convex

• Boyd & Vandenberghe, problem 3.49 (c) — proving that a function is log-concave

• Conjugate of negative normalized entropy

• Finding dual cone for a set of copositive matrices

• Boyd & Vandenberghe's pointwise supremum notation

• Boyd & Vandenberghe, example 3.11 — what's the meaning of this kind of induced norm?

• Why is this composition of concave and convex functions concave?

• Taking derivatives inside of $\inf$

• Show that the product of two convex functions $f,g$ is convex, when both $f,g$ are positive and nondecreasing

• Transforming semidefinite program into generalized eigenvalue problem (GEVP)

• How is this a linear matrix inequality?

• Is $f$ convex if the perspective of $f$ is convex?

• Pointwise Supremum of Quasi-convex Function

• Boyd & Vandenberghe, exercise 3.19(a) — I'm a bit confused

• When can I apply the step $X + t Z = X \left( I + t X^{-\frac{1}{2}} Z X^{-\frac{1}{2}} \right)$?

• Show $f(x_1,x_2) = \frac{1}{x_1x_2}$ is convex for $(x_1,x_2) \in \mathbb{R}^2_{++}$

• Boyd & Vandenberghe, section 3.3.4 — Quasiconvexity of linear-fractional composition

• Why is $xy$ not convex although it is the product of nonnegative increasing convex functions?

• Boyd & Vandenberghe, section 3.6 — Convexity with respect to generalized inequalities

• Is this proof that a closed convex function is the pointwise supremum of affine global underestimators correct?

Chapter 4

• Boyd & Vandenberghe, problem 4.6 — can one replace $=$ with $\leq$?

• Boyd & Vandenberghe, exercise 4.4(d)

• Boyd & Vandenberghe, problem 4.55 — how to show that solution is Pareto optimal?

• Why is $\inf \{ \| X \|_2 \mid X \in S^n \} = \inf\{ t \mid -t I \preceq X \preceq t I, X \in S^n\}$?

• Why isn't this function affine?

• About "optimizing over some variables" in Boyd & Vandenberghe

• https://math.stackexchange.com/q/4393502/339790

Chapter 5

• How is it a single variable optimization problem?

• How to show that the following two problems are equivalent?

• Solve optimization problem using KKT conditions

• Strong duality by Slater's condition

• Steps needed to arrive at the Lagrange dual function

• Question on mixed-strategy matrix games from Boyd & Vandenberghe

• Boyd & Vandenberghe, example 5.6 — formulation of norm approximation problem

• Problem 5.27 of Boyd & Vandenberghe's Convex Optimization

• https://math.stackexchange.com/q/4394096/339790

Chapter 6

• What is a sparsity pattern in a vector?

• Huber penalty function in linear programming form

Chapter 7

• https://math.stackexchange.com/q/4575596/339790

Chapter 8

• Why does any dual optimal point provide a separating hyperplane between a point and its projection on a convex set?

• https://math.stackexchange.com/q/4277092/339790

• Boyd & Vandenberghe, section 8.4.2 — Maximum volume ellipsoid in an intersection of ellipsoids

Chapter 9

• Why must a function be closed for descent methods to work?

• Width of an ellipsoid

• Boyd & Vandenberghe, 9.2 — how was this derived?

• Boyd & Vandenberghe, page 488 — convergence analysis of Newton's method

• Convergence analysis of quadratically convergent phase for Newton method in Boyd & Vandenberghe's Convex Optimization

• Boyd & Vandenberghe, Problem 9.6 — Gradient descent step for a quadratic function

• Gradient of a sum log function

Chapter 10

• Given $P \succeq 0$, if $x \neq 0$ and $x^T P x=0$, can we conclude $Px=0$?

• https://math.stackexchange.com/q/4692750/339790

Chapter 11

• Do primal-dual methods need to start with strictly a feasible point?

• A question about using the concavity of the logarithm in Boyd & Vandenberghe's Convex Optimization

• Why is trace operation involved when calculating gradient and Hessian?

Appendices

• What is the derivative of $\log \det X$ when $X$ is symmetric?

• Second order approximation of $\log \det X$

• Why is $\nabla \log{\det{X}} = X^{-1}$? Where did the trace go?

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To be determined

Some questions do not mention chapter, only page number. In these cases, assigning chapter to each question is more time-consuming.

• How to prove the following statement in Boyd & Vandenberghe's Convex Optimization with the help of composition rules

• Why am I getting exactly same expression for the affine underestimator?

• Question on application of randomized algorithms to solve nonconvex optimization problems

• Why is this set not a polyhedron?

• $\nabla f(x)^T(y-x) \geq 0$ if $x$ is optimal for a convex $f(x)$.

• Boundedness of sublevel sets of convex function

• How can a second-order cone problem be expressed as a conic problem?

• What is the condition number of an ellipsoid?

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Related

• What does it mean for a class of problems to have a "complete theory"?

• 1-week study plan to read chapters 1-5 of Boyd & Vandenberghe

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There should plenty more questions. If you find some, please feel free to add them to this list.