Convex Optimization is a special case of mathematical optimization. It includes Linear Programming and least-squares.

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Augmented Lagrangian Method for Inequality Constraints

Augmented Lagrangian Method can be used with inequality constraints. One approach (according to Nocedal book page 522), is linearly constrained Lagrangian. Description is shown in the attached image. ...
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Formulate an optmization problem as a convex optmization problem

Let $P$ be a polyhedron, i.e. $P = \{ x \in \mathbb{R}^{n}\, |\,\, a_{i}^{T}x \leq b_{i} \}$. Define $R$ as the rectangle given by $\{ x \in \mathbb{R}^{n}\, \mid\, \, l \preceq x \preceq u \}$. Find ...
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Finding when a function is concave [on hold]

Given the function below $f=e^{ax^2+bx}$ what are the conditions on $a$ and $b$ such that $f$ wil be concave Thanks
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1answer
39 views

intuition behind subspace of $R^n$

Hi: I've been reading an optimization text by Charles Byrne, "A First Course In Optimization". I'm currently going through the chapter where he explains things about convex sets and convex functions ...
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strong convex implies exp-concave

Prove that if f is strong convex (for some m>0) $\mbox(\nabla f(\mathbf{x})-\nabla f(\mathbf{y}))^{T}(\mathbf{x}-\mathbf{y})\geq m||\mathbf{x}-\mathbf{y}||_{2}^{2} $ then f is also ...
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1answer
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proof that length of difference of projections implies equality of length of normals

Hi: I'm reading a book on optimization and there is an interesting stated theorem but I don't know how to prove it. Notation: Let $P_{c}(x)$ denote the projection of onto a convex set c which is a ...
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Positive solutions to $A^T A x \geq 0$ [on hold]

Find a positive solution $x$ to the linear inequality $A^T A x \geq 0$. $A$ is an arbitrary matrix. I was wondering if there is a general solution. EDIT: One special solution is when $A^TA$ is row ...
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Help! convex optimization problem, how to min L1 norm and nuclear norm, like a combination of RPCA and compressed sensing. [on hold]

in the common low rank and sparse decomposition problem, in my cases I hope to replace E with Mx, I'd like to minimize the L1 norm of x rather than E.
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$\arg\max$ of an increasing function grows as the region grows.

$f_1,\dots,f_N:\mathbb{R}^+\rightarrow\mathbb{R}^+$ are strictly increasing, bounded functions whose derivatives monotonically decrease to $0$ as their argument increases. (Picture the shape of the ...
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Multivariate Optimization [closed]

I'm currently in a multivariate optimization course and literally have no idea what I'm doing. If someone could help me with this problem or at least tell me where to start I'd be eternally grateful. ...
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37 views

Non-convex function with global minimum [duplicate]

I am working on a complicated objective function which I suppose is not convex. But when I use a global optimization tool that can find all its local minimums, it will always converge to the same ...
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27 views

How can I minimize a quadratic on the unit simplex?

How can I compute $$ \min_{x \in \Delta_n} \frac{1}{2}\lVert Bx\rVert^2 + x^tAy$$ with $x \in \mathbb{R}^n, y \in \mathbb{R}^m, A_{m \times n}$, $B_{n \times n}$ where $\Delta_n$ is the unit simplex ...
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Proximal operator of spectral norm of a matrix

How can I calculate the proximal operator of spectral norm for any general matrix, $X\in R^{m\times n}$ i.e., $X^* = \arg \min_X ||X||_2 + \frac{1}{2\tau} ||X-Y||_F^2$ I understand that the proximal ...
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How can I solve $\min \{ \langle A(x),y\rangle + f(y) \text{ s.t. } y \in S^n, \operatorname{tr}(y) =1, y \geq 0\}$?

I'm trying to solve the problem $$\min \{ \langle A(x),y\rangle + f(y) \mid y \in S^m, \operatorname{tr}(y) =1, y \geq 0\}$$ where $x \in \mathbb{R}^n$, $y \in S^M$, that is, it's a symmetric $m$ by ...
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Help understanding the specification of constraints for cvxopt

This is an example from the cvxopt documentation and I am trying to understand how the L2 constraints are specified to the solver. The problem is specified as: ...
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27 views

Krein Milman Property

If a closed bounded (not compact) set $X$ in a Banach space $B$ (like $L^1$) has extreme point(s), must the max of a linear functional defined on $X$ occur at one of them? I suppose it depends on $B$. ...
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32 views

Linear Matrix Inequalities Problem [closed]

Could you help me on understanding the following problem
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How to solve this entropy optimization problem with gradient projection method?

The problem is defined as $$ \min_{w} = \sum_{i=1}^{n} \sum_{j=1}^{n}\left\{ \left(\sum_{k=1}^{3}w_kP_{ij}^{(k)}\right) \log \left(\sum_{k=1}^{3}w_kP_{ij}^{(k)}\right) \right\} + \gamma \|w\|_2^2\\ $$ ...
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Minimizing a function known to have a unique local and global minimum

Quasi-convex functions are a class of functions known to have a unique local and global minimum, which can minimized over convex sets using numerical methods with convergence guarantees. A function is ...
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24 views

Proximal operator for the nuclear norm of Hankel (x)

I have a problem in hand for which I need to compute the proximal operator of the composite function $||Hankel(x)||_{nuc}$ where $x \in R^N$ and $||.||_{nuc}$ denotes the matrix nuclear norm. For a ...
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linear approximation with respect to L1 norm

I am trying to solve this problem: Find the best $L^1$ linear approximation of $e^x$ on [0,1] i.e. minimize $\int_0^1|e^x-\alpha-\beta x| dx$ any hints how to proceed
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Find an optimal solution for $\min_{x} F(x)$ analytically

I want to find an analytical solution (exact/closed-form) for $x$ of the following minimization problem: $$\min_{x} b x \left[e^\left(\frac{a}{x}\right)-1\right]+d (1-c-x) \left[e^ ...
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How to solve the least square with $L_2$ norm constraint directly?

I answered the question Why are additional constraint and penalty term equivalent in ridge regression? earlier, but I myself still have some questions on it. To solve \begin{align} \min_{\beta} ...
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Constructing a quasiconvex function

Let $C\subset\mathbb{R}^2$ be a nonempty convex set. A function $f:C\rightarrow\mathbb{R}$ is called convex if $$ f(\lambda u+(1-\lambda)v)\leq\lambda f(u)+(1-\lambda)f(v), \quad\forall u,v\in C, ...
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Is the optimum of this problem unique?

I have a convex optimisation problem: $$\min_{x_i} \sum_{i=1}^N a_i \exp(-x_i/b_i)\quad\text{ s.t. }\sum_{i=1}^N x_i=x\text{ and } x_i\ge 0$$ The KKT conditions are: $$\lambda=\begin{cases} ...
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Reference material on Alternating Minimization Algorithm

I am looking for some good reference material (book/paper) for learning Alternating Minimization Algorithm. Any recommendation from optimization experts will be much appreciated. Thank you.
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(Updated) Geometric Illustration of Monotone and Maximal Monotone Maps

I am writing a note about the Monotone and Maximal Monotone maps from the following book http://link.springer.com/book/10.1007%2Fb97594 In this book we read a map ...
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1answer
22 views

Is this function jointly convex in its variables?

I have a function which I suspect is jointly convex, but have a difficult time proving it, especially since the Hessian is messy. The function is $f(y_i,i=1.2,\ldots,N)=\sum_i l_i w_i + y_i$, where ...
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Is there a unique saddle value for a convex/concave optimization?

Here is the question: Consider a function of two variables $f(x,y)$ on some compact domain. Let it be convex in $x$ and concave in $y$. Is it true that $f(x,y)$ has a uniqe saddle point ...
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Step-by-step example of solving a quadratic program with linear inequality constraints

I'm doing an exercise work about Support Vector Machines which involves solving a quadratic program of the form $$\begin{aligned} & \underset{\boldsymbol\alpha \in \mathbb{R}^N}{\text{minimize:}} ...
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Is the optimal solution of a strictly convex function over $\mathbb{Z}^d$ a rounded version of its optimal solution over $\mathbb{R}^d$

Consider a strictly convex function $f: \mathbb{R}^d \rightarrow \mathbb{R}$. Let $x^* = \min_{\mathbb{R}^d} f(x)$ denote the (unique) minimum of this function over $\mathbb{R}^d$. Similarly, let ...
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how to write largest circle inscribed inside a triangle as an optimization problem?

can someone show me how to write this problem as a convex optimization problem.Find the largest disk that can be bounded by $X \geq 0$ , $Y \geq0$ and $X+2Y\leq1$. My institution is to cast to ...
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Learning about convex optimisation

I'm interested in learning a bit about convex optimisation. The wikipedia article contains the following paragraph: The convexity of $f$ makes the powerful tools of convex analysis applicable. ...
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Quick Constrained Optimization Huerstistics

I am wondering if there is a way to find very quick optimization heuristics for the form. $$ f(x) = cx^a $$ $$ s.t. $$ $$ L \le Ax \le B$$ $$ 0 \le x \le \infty $$ I know with only a few variables ...
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Convexity of Conjugate Function on Determinant

If $f: \mathbb R^n \to \mathbb R$ then $conj(f) : \mathbb R^n\to \mathbb R$ is defined as $conj(f)(y) = \sup_{x \in dom_f} (y^Tx - f(x))$ and it is called the conjugate function of $f$. if $f(X) = ...
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Does every boundary segment of a convex polyhedron lies on one of its faces?

When I was reading this note, I found Theorem 3.1.5 said: Let $P\in\mathbb{R}^n$ be a polytope whose affine dimension is $d$. Then, every point on the boundary of $P$ lies in a facet of $P$. I have ...
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condition for uniform input to minimize convex function

Let $f:\mathbb{P}(X) \rightarrow \mathbb{R}$ be a convex function where $\mathbb{P}(X)$ is the set of probability distribution on (finite) set $X$. I am looking for condition on $f$ so I can said ...
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50 views

Is standard eigenvalue optimization problem convex

For any arbitrary symmetric matrix A , is the standard eigenvalue problem convex $ \lambda_{max}(A)= \max_{\|x\| \leq1} x^{T}Ax$
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Tucker's theorem from Farkas lemma

I am trying to understand the proof of Tucker's theorem using Farkas lemma but there are some points that are not clear to me. The proof I am following is in this paper at page 16. What I do not ...
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How can I solve $\inf_{x \in X} x^T(Ax + b) + \sum_{i=1}^n x_i \log(x_i)$ in a closed form depending on projection?

I'm trying to solve the minimization problem $$\inf_{x \in X} x^T(Ax + b) + \sum_{i=1}^n x_i \log(x_i)$$ where $b \in \mathbb{R}^n$, $X \subset \mathbb{R}^n$ is a convex set. And $A$ is a symmetric ...
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Are these equivalent definitions of faces of convex sets?

I consulted several books and found that the definitions of faces and exposed faces of convex sets are a bit messy. Many books just treat them as the same stuff. In book "foundations of ...
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Are these inconsistent definitions of extreme subset of convex set?

I need some elementary knowledge of polyhedral and I am consulting several books. However, I found the definitions may not be consistent. I am not sure if I understand them right. The question is ...
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Convex Optimization: Advantages of Symmetric Primal-Dual Algorithms?

This is a follow-up to an answer on a previous question on PD algorithms: http://math.stackexchange.com/a/1193928/36257 I have done some research learning the mechanics of how Infeasible PD ...
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Is $\mathbf{y}^*$ a local minimizer of $f(\mathbf{h}(\mathbf{y}))$?

Let $f(\mathbf{x})$ be a twice differentiable function, where $\mathbf{x} \in \mathbb{R}^n$. Let $\mathbf{x}^*$ be a local minimizer of $f(\mathbf{x})$. Consider a differentiable and invertible ...
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How can I solve $y \in (N_X + \nabla f)(x)$ via projection?

I a aware that if I'm trying to solve for $x$ the problem $y \in [\lambda I + N_X](x)$ where $y$ is a known vector, and $N_X$ is the normal cone given by $N_X(x) = \{u : \langle u, x - y\rangle ...
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How to use CVX to solve this problem?

I have a function in the variables $x_{kl};\ k,l=1\ldots,m$, $$\sum_{i=1}^n \sum_{j=1,j<j'}^{N_i}\left( b_{ij} b_{ij'}- \sum_{k,l=1}^{m}x_{kl}f_k(a_{ij})f_l(a_{ij'})\right)^2$$ where ...
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Minimizer of $\frac{\lambda}{2} \| \theta - \theta^{(k)}\| + \text{Loss}_\text{hinge}(y \theta \cdot x)$

How do you find the minimizer of: $$\min_{\theta \in \mathbb{R}^d}\left\{ \frac{\lambda}{2} \| \theta - \theta^{(k)}\|^2 + \text{Loss}_\text{hinge}(y \theta \cdot x)\right\}$$ if $\theta^{(k)} \in ...
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SVM Soft Margin Lagrange form

I study the Lagrange multipliers form of SVM. I am particulary interested in values that $\alpha_i$ can get. The following is the Langange multipliers form of hard margin SVM. $min_{w,b} ...
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Minimization over a symmetric matrix

I'd like to know what are possible methods to minimize over a symetric matrix R. Example: min $||AX -B||_2^2$ The minimization is over A, such that $A^T = A$, $A \in R^{3x3}$, $X \in R^{3x\alpha}$, ...
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Affine hull example

In the textbook "Convex Optimization", S. Boyd says that the affine hull of a set $C\subseteq \mathbb{R}^{^{n}}$ is the smallest affine set that contains C. Moreover, the Ex. 2.2 shows the set $ ...