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Questions tagged [convex-optimization]

A convex optimization problem consists of either minimizing a convex objective or maximizing a concave objective over a convex feasible region.

Convex Optimization is a special case of mathematical optimization where the feasible region is convex and the objective is to either minimize a convex function or maximize a concave function. Linear Programming is a special case. Convex Optimization problems as a class are easier to solve numerically than general mathematical optimization problems.

The following problems are all convex minimization problems, or can be transformed into convex minimizations problems via a change of variables:

  • Least squares
  • Linear programming
  • Convex quadratic minimization with linear constraints
  • Quadratic minimization with convex quadratic constraints
  • Conic optimization
  • Geometric programming
  • Second order cone programming
  • Semidefinite programming
  • Entropy maximization with appropriate constraints
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Difference between supremum and maximum

Referring to this lecture , I want to know what is the difference between supremum and maximum. It looks same as far as the lecture is concerned when it explains pointwise supremum and pointwise maximum
user31820
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Why concept of proper cone is important in convex optimization?

I'm reading Convex Optimization from Prof. Boyd and I got some question on the important usage of proper cone in convex optimization. Other than generalized inequalities, is there any other use of proper cone in convex optimization?
MakaraPr
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What is an example of non-convex cone

I was studying Stephen Boyd's textbook on convex optimization. It says "A set C is called a cone or nonnegative homogeneous, if for every x $\in$ C, we have $\theta x \in $ C. A set C is a convex cone if it is convex and a cone." I'm just wondering…
DSKim
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L1 norm and L2 norm

I was studying the Stephen Boyd's textbook on convex optimization. It says the following: The amplitude distribution of the optimal residual for the l1-norm approximation problem will tend to have more zero and very small residuals , compared to the…
DSKim
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Convexity of log det X???

In Boyd's book on convex optimization he proves convexity of log det X by proving it to be concave along a line i.e. he proves that the Hessian of the function $g(t) = f(Z+tV)$ is negative therefore this function is concave. He assumes that $Z \in…
Shirin
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The difference between affine set and affine hull

According to the definition of affine hull and affine set. $$aff [C] = [\theta_1x_1+...+\theta_nx_n|x_1,...x_n \in C, \theta_1+...+\theta_n=1] $$ The data in affine hull is also in affine set. And vice versus. So what's difference between them?
Samuel
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For a convex optimization problem, if primal and dual optimal variables exist, does strong duality hold?

Is there a convex optimization problem such that a primal optimal solution exists, and a dual optimal solution exists, but the primal optimal value is not equal to the dual optimal value?
eternalGoldenBraid
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Why is any subspace a convex cone?

I am reading Convex Optimization written by Stephen Boyd. In page 27 of chapter 2, there is an example said 'Any subspace is affine, and a convex cone(hence convex).' Can anybody explain to me why this is true?
BioCoder
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Boundedness of sublevel sets of convex function

(This is from page 474 of Boyd & Vandenberghe's Convex Optimization, on algorithms for unconstrained minimization) Assumptions The function $f : \mathbb{R}^N \mapsto \mathbb{R}$ is convex and twice-differentiable and there exists an optimal point…
buzzie13
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Convex Hull definition and counterexample?

The convex hull of a set $C$ is $$conv(C) = \{\theta_1x_1 + \theta_2x_2 + \cdots + \theta_k x_k: x_i \in C,\theta_i \ge 0, \sum \theta_i = 1, k=1, 2, \ldots \}$$ I am wondering why it is not enough to take $k=2$ in the definition. Is there a simple…
Fequish
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Logistic function as "difference of convex functions" (DC)

is there a way to express the logistic function $$\frac{1}{1+\exp(-x)}$$ as the difference of two convex functions? Thanks
TheDon
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Finding a feasible point under quadratic constraints

Lets say we have a Quadratically Constrained Quadratic Program which we would like to optimize. The first step for many methods requires one to first find a point in the feasible region. How can I answer the question "Is the feasible region…
Tim Seguine
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Understanding the conjugate of a function

I am bad at math and am having an extremely hard time trying to understand the conjugate of a function. I am not sure why the dotted line is the conjugate function. How do I find that? This is my limited understanding of the steps. 1) draw $y^Tx$…
Kong
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The primal problem has a unique solution while the dual problem has many solutions

Is it possible that the primal problem has a unique solution while the dual problem has many solutions? If so, does this situation have a particular name?
Angelos
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Minimizing a sum of powers

For positive integers $ b_1,b_2,...,b_T $, I am trying to solve the following optimization problem: $$ \min_{\substack{p_1,\ldots,p_T\\ p_1+\cdots+p_T=1\\}} \sum_{i=1}^T {p_i} ^ {b_i} .$$ The solution I would like to find is an expression for $ p_i…
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