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

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Confusion related to optimization of log(det(X)) function

I have this confusion related to optimization of the log(det(X)) function. I didn't get how it implicitly maintains the constraint of X being positive definite. For eg if I have a matrix ...
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97 views

why in Phase I of the simplex method, if artificial variable become nonbasic, it never become basic?

Does anybody has idea how to solve this problem ? "Show that in Phase I of the simplex method, if an arti cial variable becomes nonbasic, it need never again become basic. Thus, when an arti cial ...
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54 views

Suggestions for a reference-level text on optimization theory?

I'd be interested in knowing if anybody has suggestions on an advanced but still self-contained reference on optimization theory, centered around linear and convex problems. The key feature of my ...
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103 views

non-degenerate basic feasible of Polyhydron

I couldn't show this problem. Can somebody help me by this question? Consider a polyhedron $\{X \in \mathbb{R}^n | AX \leq b, X \geq 0 \}$ and a non-degenerate basic feasible solution $X^*$. We ...
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31 views

“Buzzword” for approximate gradients (that form a positive scalar product with the real gradient)

Let $\vec g(\vec x)\in\mathbb R^N$ be the gradient of a convex function $L: \mathbb R^N\mapsto \mathbb R$ and $\vec h(\vec x)$ such that $$ \vec h(\vec x)^T\vec g(\vec x) \geq 0\quad\quad \forall \vec ...
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50 views

Solution of a Quadratic Optimization Problem

Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two given $N\times N$ hermitian matrices. Then how do I solve the problem, \begin{align} ...
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22 views

I need a resource for basic convex optimization algorithms.

I'm trying to decide whether or not a certain CS problem can be solved in polynomial time. I've got it reduced down to a basic convex optimization problem, but I can't for the life of me find a good ...
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34 views

Basic questions about convex optimization

I have some basic questions about convex optimization. From finding sources online, I've seen that many algorithms (for example, Newton's method) describe themselves as $o(\frac{1}{\epsilon})$. ...
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19 views

Which methods of function continuation admit polynomial-time convex minimization?

The function $f$ maps the positive integer lattice in $\mathbb{R}^n$ (i.e. the vectors in $\mathbb{R}^n$ whose coordinates are all integers) to $\mathbb{R}$. We know that $f$ is convex. I want to ...
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28 views

A question about monotonicity

Is $$D(y_l)=\int_{-\infty}^{y_l}f_0(y)\mbox{d}y+\int_{y_l}^{y_u}e^{x\ln(1/L(y_l))}L(y)^{x}f_0(y)\mbox{d}y+\frac{1}{L(y_l)}\int_{y_u}^{\infty}f_0(y)\mbox{d}y$$ with ...
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140 views

Facets of the convex hull as solution of an optimization problem?

Given $N$ points $x_1, x_2, ..., x_N \in \mathbb{R}^n$, consider their convex hull $$\mathcal{C} = \text{conv}( \{ x_1, ..., x_n \} ) = \bigcap_{j=1}^{J} \{ x \in \mathbb{R}^n : \ A_j x \leq b_j \} ...
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83 views

Formulation of a problem as semidefinite programming

I would appreciate some help with this problem: $R$ is a positive semidefinite matrix $\in{R}^{n\times n}$, $A \in{R}^{n\times m}$. I need to formulate this optimization problem as semidefinite ...
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156 views

About Schur complement in a non-linear matrix inequality

I have the following matrix inequality which is nonlinear due to $M^TM$. In order to transform into an LMI, I apply the Schur complement, however I am not sure about the result. Can you tell me if ...
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53 views

Is it problematic when using Newton Descent with discontinuous Hessian?

Is there any side effect when applying Newton Descent to a convex function whose Hessian is discontinuous?
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142 views

Maximizing an inner-product over a convex set.

Let $x \in \mathbb{R}^N$ and let $K$ be a closed convex set in $\mathbb{R}^n$. Let $$ \widehat{y} = \textrm{arg} \, \textrm{max} _{\,\,y \in K} \langle x,y \rangle,$$ where $\langle \cdot, \cdot ...
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31 views

Confusion related to convexity and concavity of a problem

I was reading this paper http://www.ist.temple.edu/~vucetic/documents/wang11kdd.pdf related to adaptive multi-hyperplane machine for non linear classification In that paper, they have mentioned about ...
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24 views

Can we express a SPD matrix $S$ in terms of $S^{2}$ in a different manner to solve a convex problem?

I have to find the Symmetric Positive Definite matrix $S\in \mathcal{M}_{m,m}$ that minimizes the function $f(S)=\mathrm{trace}(S)+\mathrm{trace}(S^{-2})$ which has been proven to be convex in the ...
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110 views

maximize an objective function with an infinite component

Suppose I have the following maximization problem: $\log\det(\alpha K_p)-c\alpha$ with respect to $\alpha$ with $c$ being a constant and $m$ being the dimension of $K_p$. Here, one of the eigenvalues ...
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146 views

K.K.T. conditions, Lagrangian gradient not defined for zero.

When I write the K.K.T. conditions for the problem I have, I get the following expression for the gradient of the Lagrangian: $$\frac{\partial \mathcal{L}}{\partial x} = - \frac{\sqrt{x} + ...
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195 views

How to find $\kappa$ to minimize integral $I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) \mathrm{d}x$

I am trying to find such value $\kappa \in (0,1)$ that would minimize the integral \begin{equation} \begin{aligned} I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) ...
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91 views

Minimize a complex quadratic subject to two convex quadratic constraints

I have the following the optimization problem \begin{align} \min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~&||\mathbb{x}^H\mathbb{u}||_2^2-2*Real\{ \mathbb{x^Hu}\} \\\ ...
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60 views

Parameterized convex optimization

I'm trying to formulate a game so that at Nash equilibrium I achieve supply equales demand. Then I ran into this problem. For all $i,$ $v_{i}\left(x_{i}\right)$ is concave in $x_{i}$. The value ...
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59 views

Closed form for Lagrange dual

Can Lagrange dual always be computed in closed form? Can you give me a simple example where the dual is not analytically computable?
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50 views

why is it important to have $\max_x \min_y f(x,y)=\min_y \max_x f(x,y)$?

I am currently trying to understand the minimax theorem of Von Neumann and the improved versions of this theorem. At any case we have the property $$\max_{x\in X} \min_{y\in Y}f(x,y)=\min_{y\in Y} ...
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255 views

Linear programming: writing a problem with artificial variables?

Use artificial variables to write a linear programming problem in canonical form with non-negative resource vector whose solution will determine whether there exists (and if so, find) non-negative ...
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47 views

Duality gap in cone programming

Let $K\subset \mathbb{R}^2$ be a closed convex and pointed cone, $A$ be a $2\times 2$ square matrix and $b, c\in \mathbb{R}^2$. Consider the problem $$ (P)\quad \min\{\langle c, x\rangle: Ax\geq_K ...
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47 views

General properties of an optimal solution of a convex program

How do we seek certain properties for a solution of a convex minimization problem. For example we want to make sure if the below objective has a symmetric optimal solution: \begin{equation} \min_X ...
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37 views

using the ellipsoid algorithm to find a poly time algorithm for the optimization problem

Consider the following optimization problem: Let $n$ be even and let $c, x$ be positive vectors in $\mathbb{R}^n.$ Find $\min(c^Tx)$ for $\sum_S x_i\geq 1,$ for any $S\subset \{1,...,n\}$ with $|S| ...
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131 views

Upper bound for L1-L2 optimization problem

I am interested in the following convex optimization problem: \begin{align*} \max & ||x||_1 \\ \text{s.t.} & ||x-a||_1 \le K \\ & ||b\circ x||_2 \le 1\\ & x \in R^n \end{align*} where ...
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94 views

minimum of the function over symmetric body

Let $X$ be a normed space. Let $K$ be centrally symmetric convex body with $M$ known vertices and with $Vol(K)=V$. We subdivide body $K$ be $M$ equal pieces, $P_i, i=1, \ldots, M$, such that each ...
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164 views

Efficient Algorithm For Projection Onto A Convex Set

Given $\mathbf{x} \in \mathbb{R}^n$ and $\tau$ a scalar, I would like to solve the following Euclidean projection problem: $\underset{p}{\mathrm{argmin}} \; \|\mathbf{p}-\mathbf{x}\|_2 \;\; ...
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240 views

A convex programming problem involving sum of logarithms of linear functions

Here is a convex programming problem I encountered while working on an estimation problem for a mixture of multinomial distributions. We have a matrix $A_{m \times n}$ containing non-negative real ...
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133 views

How to calculate the maximal ellipsoid in a given polyhedron

I am faced with the problem of finding the ellipsoid $B$ ($B$ is a symmetric positive definite matrix) of maximal volume within a convex set $C$ given as a set of linear inequalities $C=\{x| a_i^T x ...
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170 views

conjugate function for matrix mixed norm

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows: ...
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289 views

Is conditional entropy a convex function?

A conditional entropy can be expressed in the following way, $H_{V_t}(V_s) = -\sum_{s,t}p(s,t)\log{p_t(s)} = -\sum_{s,t}p(s,t)\log{\frac{p(s,t)}{\sum_{s'}{p(s',t)}}}$ $s$ and $t$ are defined ...
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51 views

Is it possible to solve the argument maximization problem $\arg\max_x \langle x,l\rangle - f_1(x) - f_2(x)$ via convex duality?

I am attemping to solve the argument maximization problem $$\arg\sup_x \{\langle x,l\rangle - f_1(x) - f_2(x)\}\qquad\qquad\qquad\qquad (1)$$ where the functions $f_1$ and $f_2$ are concave but ...
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166 views

Convexity of a Set

Consider the following function, $$ f(x, y) = e^{m e^{-y}+n e^{-x}-x-y} \left(a x e^y+b e^x y+c x y\right) $$ where $a, b, c, m$ and $n$ are positive constants. I want to show $f(x, y)$ is ...
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8 views

online algorithm for nested optimization

How to construct a sequence {x_t;\theta_t}, which is online algorithm for following optimization problem: $\arg\min_\theta \sum_t \min_{x_t} \ell_t(x_t;\theta)$ For simply, we can assume ...
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29 views

Linear Programming - Tableau Condition

The following tableau corresponds to an iteration of the simplex method: ...
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21 views

Convex sets and minimum points

Let $X$ be the convex set formed by the convex combination of the $n$ points $\{x_1, x_2, ... x_n\}$ in $\mathbb{R}^n$. Let $X^* \subseteq X$ be the convex set of minimal points w.r.t to the convex ...
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18 views

A Difficult combinatorial optimization problem

Let $\mathcal{J}$ be a closed, bounded, compact, convex set in $\mathbb{R}^L$. (Notations: vector $\mathbf{x}$ is denoted in bold letters and its $i^{th}$ co-ordinate is denoted as $x_i$. ...
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14 views

Eliminating variables in convex program

This is a basic convex optimisation question. I have the following problem: $$\max_{\substack{t\le e\\ At\le b}} e^\top t$$ How do I find the optimum $t^*$? I write the KKT conditions, get ...
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32 views

weights go to infinity in logistic regression with linearly separable data

I have the loss function of logistic regression $L(W)$ = - $\sum_{i=1}^n {y_i}.log[\sigma(w^Tx)] + {(1-y_i)}.log[1- \sigma(w^Tx)]$ I have derived the Hessian and proven it's positive semi-definite ...
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10 views

A strange sufficient condition for quasiconcavity

I think I learned this from a lecture today If a multivariate C2 function is increasing and its bordered Hessian has positive determinant, then f is quasiconcave. ...
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5 views

Verification of the Approach to a given non-convex integer programming problem

I need to verify my approach to a non-convex integer programming problem. It would be interesting to see other approaches as well. Let $\mathbf{C}_1,\dots,\mathbf{C}_R$ be $N\times N$ hermitian ...
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10 views

Quasiconcave condition for a power function

Let $f(x, y)= (ax^2+by^2)^n$ where $a, b, n$ are positive, $x, y\in \mathbb{R}$. What is the condition of $n$ so that $f(x, y)$ is a quasiconcave, and concave function? My idea is only calculate ...
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27 views

Least squares and simplex

I am interested in the linear least square problem with the solution with the following constraints : $$ \min_x \|Ax-b\|^2$$ subject to $0 \le x_i \le 1$ and $\Sigma_{i=1}^n x_i= 1$. Because of the ...
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16 views

Let $P$ be a polyhedron. Prove, $P$ has at least one extreme point $\iff$ $P$ does not contain a line, by using a lemma.

Let $P$ be a polyhedron. Prove, $P$ has at least one extreme point $\iff$ $P$ does not contain a line, by using a lemma. I've a Lemma saying: Suppose $P=P(V,E)$ where $V,E \in \mathbb R^n$ are ...
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12 views

Let $P \subseteq R^n$ be a polyhedron. Why does $\{ x + \alpha d \mid \alpha > 0\} \subseteq P$ for some $x \in P$ imply $d$ is a recession direction?

Suppose we have a polyhedron $P \subseteq R^n$ and let $d \in P$ be a recession direction, that is $\{ x + \alpha d \mid \alpha > 0\} \subseteq P$ for all $x \in P$. Why does $\{ x + \alpha d \mid ...
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20 views

Are iterations involving quantization going to converge?

For $i = 1,2,3$, let $~f_i(y_i)~$ be a convex and differentiable function and $y_i$ a scalar variable. Consider the following iteration $$\left[ \begin{array}{c} \nabla f_1(y_1^{k+1}) \\ \nabla ...