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

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Why is this function, related to SVM derivation, non-convex?

I'm working through a support vector machines tutorial. In eventually deriving the solvable objective function, the following objective function (to be maximized) was proposed, but dismissed as ...
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59 views

Is there any way to make the following function convex?

I need to find optimal lagrangian multiplier vectors for a quadratic programming problem subject to three quadratic equality constraints and several other linear inequality constraints. I would like ...
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52 views

How to project gradient vector to subspace defined by linear constraints

I have the following set of linear constraints: $$\begin {align}\textbf{y}^T\textbf {x} &= 0 \\ \textbf {0} &\leq\textbf {x} \leq C\cdot\textbf {1},\end {align}$$ where $\textbf {y} \in ...
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21 views

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

I realize that a second-order cone is a cone, and thus an SOCP is a type of conic problem. However, to me it doesn't seem so apparent, looking at their equations. Could someone explain how one could ...
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24 views

How can I compute fast the minimum of a linear plus Kulback-Leibler on the unit simplex?

Given $a, x^0 \in \mathbb{R}^n$ I wish to compute $$\min_{x \in \Delta_n} a^t x + \sum_{i=1}^n x_i\log(x_i/x^0_i) - x_i +x^0_i $$ where $\Delta_n$ is the unit simplex $\{x \in \mathbb{R}^n \mid ...
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30 views

Convergence rate - Convex optimization

What is the best known algorithm in terms of convergence rate for unconstrained convex optimization and under what assumptions? $\min_{x} f(x)$ where $f(x)$ is a given twice differentiable convex ...
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20 views

Show the Gini Coefficient is Quasiconvex

The Gini-coefficient is defined as $$ G(x) = \sum_{i = 1}^n \frac{i}{n} - \sum_{j=1}^{i} \frac{x_{(j)}}{\mathbb{1}^{T}x}, $$ where $x_{i} $ is nonnegative numbers with positive sum. $x_{(j)}$ denotes ...
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38 views

Augmented Lagrangian Method for Inequality Constraints

Augmented Lagrangian Method can be used with inequality constraints. The question is how. One approach (according to Numerical Optimization Book by Nocedal and Wright; page 522), is linearly ...
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44 views

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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46 views

Convex Optimization: do Primal Dual methods need to start with strictly feasible point?

I'm learning about Primal-Dual interior point algorithms from Boyd & Vandenberghe Convex Optimization, ch.11.7. Now the text mentions: In a primal-dual interior-point method, the primal and ...
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33 views

Proving a function of matrix is convex

I have a function of a matrix and a vector $f(A,b)=y^\top (I-A)^{-1} b$ and I want to know the conditions under which it is convex. For functions of a vector, the positive definiteness of the Hessian ...
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51 views

How to derive the solution in quadratic optimization

I'm reading the book "Convex Analysis and Optimization" written by Prof. Bertsekas. In Example 2.2.1, there are the following description: I don't know how to ...
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20 views

Show the following statements are equivalent - convexity

Let $C \subset \mathbb{R}^n$ be a set. Show the following are equivalent: (a) The set $C$ is convex. (b) The function $\delta_C : \mathbb{R}^n \to \mathbb{R} \cup \infty$ defined as: ...
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36 views

Convexity proof - can I get some pointers?

Prove that $C \subset \mathbb{R}^n$ is convex iff $\forall m \in \mathbb{N}$ and every set of $m$ points $\{x_1,...,x_m\} \subset C$ we have that $\sum_{i=1}^m \lambda_i x_i \in C$ Where ...
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34 views

how to find the solution of this cost function?

I have the following cost function. $J = \sum_{i=1}^N a\, Trace(W^TX_iW) - b\, Trace(W^TY_iW)$ Where $X_i$ and $Y_i$ are symmetric matrices, $a$ and $b$ are scalars. How can I find W?
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18 views

Convexity of a subset is convex?

Let $V$ be the set of sequences whose terms are contained in $\mathbb{R}^n . V$ is the set of functions $x(·) : N → \mathbb{R}^n $ which we denote as $\{x_n\}_n \subset \mathbb{R}^n$. $V$ is a vector ...
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36 views

Showing affinity of a function - proof help

Let $V$ be the set of sequences whose terms are contained in $\mathbb{R}^n . V$ is the set of functions $x(·) : N → \mathbb{R}^n $ which we denote as $\{x_n\}_n \subset \mathbb{R}^n$. $V$ is a vector ...
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46 views

Help with this convex set proof

Take $C ⊂ \mathbb{R}^n$ a convex set. Fix $x_0 ∈ C$ and a nonzero vector $v ∈ \mathbb{R}^n$ . Define the set $I(x_0,v) := \{t ∈ R : x_0 + tv ∈ C \}$. Prove that $I_(x_0,v)$ is a convex subset of ...
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78 views

infimum and supremum notation

I have stumbled across this blob of text when reading my textbook, and would like to know how to interpret it more intuitively. I understand the definitions of inf and sup, however not so much what ...
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51 views

Showing the intersection/union of a cone is a cone

Defining a set $C \subset \mathbb{R}^n$ as a cone if for ever $x \in C$ and $\alpha \geq 0$ we have $\alpha x \in C$. ie they are closed under scalar multiplication. How can I show that the ...
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44 views

How can I show the following statements are equivalent?

Let $C ⊂ \mathbb{R}^n$ Prove that the following statements are equivalent. (i) $C$ is an affine set (ii) For every $x_0 ∈ C$ , the set $C − x_0 := \{ z − x_0: z ∈ C \}$ is a subspace. (iii) There ...
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55 views

Showing convexity proof

Let $F: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be an affine function, i.e., $F (x) = L(x) + b$, with $L : \mathbb{R}^n \rightarrow \mathbb{R}^m$ linear and $b \in \mathbb{R}^m$ Then for every convex ...
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132 views

Scale ellipsoid maximally within polyhedron

Given an ellipsoid around the origin with scaling parameter $e$ in the form $x^T E x \leq e$ and a polyhedron $P$ given by $A x \leq b$, how can we define an optimization problem that maximizes e such ...
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19 views

Conjectured characterization of a set relative to a convex cone

Let $X\subset \mathbb{R}^N$ be a convex cone (i.e., for all $x,y\in X$ and $\alpha,\beta\geq 0$ scalars, $\alpha x+\beta y\in X$). Define the set $$A(x)=\{a:x+a\in X \wedge x-a\in X\}.$$ Then, ...
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39 views

Efficient solution for a quadratic + norm objective.

I want to minimize an objective function of the following form: $$ \begin{split} \text{Minimize} \quad & x^T D_x x + y^T D_y y + z^T D_z z + q_x^T x + q_y^T y + ...
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43 views

What does 'the level set is bounded' exactly want to tell?

'The level set is bounded.' occurs in many theorems and other places. I think I can understand the definition of 'level set' but I don't know what does 'it's bounded' want to tell me exactly in ...
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2answers
92 views

Local optimality of a KKT point.

Consider the problem \begin{equation} \min_x f(x)~~~{\rm s.t.}~~~ g_i(x)\leq 0,~~i=1,\dots,I, \end{equation} where $x$ is the optimization parameter vector, $f(x)$ is the objective function and ...
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32 views

Express this linear optimization problem subject to a circular disk as a semidefinite problem.

I have to express following problem as a semidefinite problem: $ min \, F(x,y) = x + y +1$ subject to (1) $(x,y) \in \mathbb{R}^2 : (x-1)^2+y^2\leq 1$ Only affin equality conditions should be used. ...
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25 views

Is it right for chain rule in trace function?

The objective function is $$ f(X)=\min_X trace(B^TX^TCXBD) $$ we know the following derivatives from Matrix Cookbook, $$ \frac{\delta{trace(B^TX^TCXB)}}{\delta X}=C^TXBB^T+CXBB^T \\ \frac{\delta ...
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50 views

Optimization with changing objective function

Is there any theory about (convex) optimization where the objective function is allowed to change during the optimization process? I have a problem where the objective function depends on some ...
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54 views

Why does gradient descent make sense?

Suppose I define two functions of $x$ in terms of a convex function $f$ with a unique minimum $x_0$: $$f_1(x) = 1 \times f(x)$$ $$f_2(x) = 2 \times f(x)$$ Suppose I wanted to minimize each of these ...
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35 views

Optimisation over matrix entries

I was looking to write the KKT conditions to solve this optimisation problem. $$\min_{\substack{\sum_j x_{ij}\le k_i \\ i=1,2,\ldots N}} a^\top (I-X)^{-1} b $$ Since there are $N^2$ decision ...
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70 views

Minimization over two lines

This is a minimization question where the minimizing points can be chosen freely on two lines: $$\mbox{minimize}\, \prod_{i=1}^K {y_i}\quad \mbox{such that}\quad \prod_{i=1}^K ...
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51 views

Dual Optimization Problem

I have the following optimization problem, $$ \text{minimize}_{X,Y} \ \lVert X\rVert_* + \lambda \lVert Y\rVert_1 \\ \text{subject to }X + Y = C$$ $C \in \mathcal{R}^{m \times n}$, $\lVert ...
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68 views

Matrix Maximization

I would like to solve the following optimization problem for a matrix $X$ which is symmetric and positive-semidefinite: $$ \mathrm{maximize} \, \, \, f(X) = \log \mathrm{det} X - k_1 \log(k_2 + a^T X ...
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191 views

Why is L21 norm not smooth

I have this confusion. I was reading this paper http://www.cis.temple.edu/~yuhong/research/papers/ijcai13b.pdf. I didn't understand why is L21 norm not smooth?
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59 views

A particular quadratic minimization problem

Given $n^2$ constants $a_{11},a_{12},\ldots,a_{1n},a_{21},\ldots,a_{nn}$ and $n^2$ non-negative variables $x_{11},x_{12},\ldots,x_{1n},x_{21},\ldots,x_{nn}$. Find the minimum value of $$\sum_{i=1}^n ...
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51 views

Linear constraint in convex optimization

Is it true that the solution to a linearly constrained convex minimization problem can only be placed on the boundary of the constraint set, for any nonlinear convex objective, e.g. $$ \min_x f(x)$$ ...
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37 views

Problem understanding dual optimization problem?

I am reading this paper: http://dl.acm.org/citation.cfm?id=1390696 Following optimization problem is defined in section 2: \begin{align} \max_{\mathbf{X}>0} \log ...
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2answers
161 views

Minimization of log-sum-exponential function subject to constraints.

I would like to minimize the following function: $f(x)=log(e^{-x_1}+..+e^{-x_n})$ Subject to: $\sum_{i=1}^{n}{x_i}=1$ $0 \leq x_i \leq 1$ So far I have discovered the following: If all the ...
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46 views

Quadratic programs: is the projection onto constraints optimal?

Consider the Quadratic Program $$ x^* := \arg \min_{ x \in X } \ \{ x^\top x + c^\top x \} \ \text{ sub. to: } Ax=b $$ where $X \subset \mathbb{R}^n $ is a non-empty, convex, bounded polyhedron. ...
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42 views

Projection onto convex set defined by $\|\mathbf{t} -\mathbf{W}^T\mathbf{y}\|^2 \leq k$

I want to use the method of Projections Onto Convex Sets, and for the problem at hand I need to find a closed form solution for $\mathbf{P}_C$, the projection onto set $C$, defined as: $$C = \{ ...
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37 views

Is a constrained optimization problem equalivant to its Lagrangian form?

For the following problem: $\text{min:}\ f(x)\\ s.t. \ g(x)\leq t$ Is the above problem equalivant to the following problem? $\text{min:}\ f(x) + \lambda g(x) \\ s.t. \ \lambda\geq0$ where $t$ and ...
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57 views

Convex Sets and extreme supports

Let the set $S$ in $R^n$ consists of the origin $0$ and $n$ lineary independent vectors $T_1, \ldots, T_n$. Show that $C(S)$, the convex hull of of $S$, is the intersection of its extreme supports, ...
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40 views

Derivative of a minimum

The expression, $e=\left(x(t,w)-c_x\right){}^2+\left(y(t,w)-c_y\right){}^2$, has a local minimum with respect to $t$ at some $t_0(w)$. Now what does $t_0'(w)$ look like?! $x,y\in C^2$ with respect to ...
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40 views

Impact of constraints on convex optimization

Let me start by saying I know almost nothing about optimization so please bear with me. Basically, I am wondering whether it is possible to solve a problem with two constraints by solving the problem ...
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46 views

An application of Separation Theorem

Let $X$ be a Hausdorff locally convex topological vector space. Suppose $X_0 \subset X$ is nonempty convex set, $g:\; X\to \mathbb R^m$ is a convex vector function (each component $g_i(x): X\to ...
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36 views

Propertise of a Dual Cone

In Convex Optimization by Boyd (P.51) said that " $y\in K^*$ iff $-y$ is the normal of hyperplane that supports $k$ at the origin ($K^*$ is a dual cone of $K$) " what does it mean geometrically? I ...
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59 views

Describing a Dual Cone

1)Does dual cone define just for proper cone or all kinds of cone ? 2)Can someone show me a figure that shows a dual cone of a cone ? In Convex Optimization by Boyd (P.51) said that " $y\in k^*$ iff ...
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106 views

Partial derivative on convex set

If we have a function $f:U \rightarrow R$ ($U \subset R^n$) which is partially differentiable on a convex set U with $\frac{\delta f}{\delta x_1} = 0$ for all $x \in U$. How can we prove that $f$ ...