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.

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How to find max and min bounds of a uncertain function

First I would like to say that I have searched the for uncertain fitting, robust fitting, linear optimization, convex optimization, etc. But I'm lacking the knowledge to solve this problem, and I need ...
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2answers
74 views

Numerical minimization of scalar-valued function in 3d

I am finding minimum of the potential function $f=f_1+f_2$, where $f_i: \mathbb{R}^3\to\mathbb{R}$. I was about to use Levenberg-Marquardt as the quick starting point, since it is already implemented ...
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1answer
52 views

How can I convert this SDP constraint?

I have the following SDP problem: max: $Tr(CX)$ subject to: $X \geq 0, I - X \geq 0$. I want to convert it into the standard form specified by CSDP (I'm using the callable C interface), which is: ...
3
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2answers
83 views

A separable nonnegative quadratic program

I have spent quite some time trying to solve the following quadratic program: $$\min \sum_{i=1}^n (\frac{1}{2}x_i^TQx_i+c_i^Tx_i), \quad \mathrm{s.t. } \quad x_i\ge 0 \quad \forall i,$$ where $n$ is ...
3
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2answers
86 views

Non-linear root finding with positive definite Jacobian

I am dealing with a system of non-linear equations: $$ f(\boldsymbol{x}) = \boldsymbol{y}, \;\;\; \boldsymbol{x}, \boldsymbol{y} \in \mathbb{R}^d. $$ And I know that the Jacobian $J(\boldsymbol{x})$ ...
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2answers
94 views

Matlab fmincon with zero user-supplied hessian

I have to solve the problem $$\min_x 1^Tx+\frac{\lambda}{2}\|\Omega\mu-x\|_2^2+\frac{\beta}{2}\|x-\bar{\gamma}\|_2^2\quad\text{w.r.t.}\quad Px-c=0,\ \ x\geq0$$ and in order to do that with Matlab I ...
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0answers
61 views

About Convex Geometry

A consistency notion in constraint programming: Let $P = (X, D, C)$ be a CSP. Given a set of variables $Y \subseteq X$ with $|Y| = k -1$, a locally consistent instantiation $I$ on $Y$ is ...
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1answer
145 views

Matrix completion algorithm

I am trying to implement the algorithm presented in this paper which tries to recover a matrix that represent a less noisier dataset of the intensities of the pixels of a set of images. In this case ...
0
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1answer
118 views

CPLEX claims to have solved QP minimisation but solution is not optimal

I am trying to solve a small QP problem in CPLEX. The problem has several linear constraints. The optimiser runs and finds a solution which satisfies these constraints and CPLEX returns a success ...
0
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1answer
49 views

Elastic LP Programming

Say I have an LP that is unfeasible and that I want to find the solution that makes it feasible without strongly violating the current constraints. What is a principled way of solving this problem, ...
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3answers
98 views

Plane constraints in R3

I have multiple plane constraints in $\mathbb{R}^3$ of the form: $$n_i \cdot x \ge \delta_i$$ Where $n_i$ is the $i$th plane normal (in form (x, y, z)), $x$ is a point in space, and $\delta_i$ is ...
0
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1answer
75 views

Custom CVX Functions - Overriding Errors [closed]

I have constructed a function in Matlab that is convex and increasing (qualitatively similar to an exponential function but I am hoping to avoid the successive approximation requirement of exp). In my ...
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1answer
123 views

Non-linear optimization using approximate gradient

I'm working with non-linear optimization for imaging, such as MRI and CT. Our problem is of the form $\|Ax-b \|_2^2+\lambda \|Wx\|_1$. $A$ is never formed explicitly, so we're limited to approaches ...
5
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2answers
126 views

Minimizing 1D convex functions

I have a one dimensional convex function $$f : [a,b] \to \mathbb{R}$$ and want to find the minimum value $$\min_{a \le x \le b} f(x)$$ I know all derivatives of $f$, so the problem could easily be ...
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4answers
373 views

Linear programming with matrix constraints

I have an optimization problem that looks like the following $$ \begin{array}{rl} \min_{J,B} & \sum_{ij} |J_{ij}|\\ \textrm{s.t.} & MJ + BY =X \end{array} $$ Here, my variables are matrices ...
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1answer
69 views

Convex objective function of matrix with prescribed determinant and trace

I have real symmetric positive definite matrix $M = \left(\begin{matrix} a & b \\ b & c \end{matrix}\right)$ where $a,b,c \in R,\ a,c>0,\ \left|b\right|<2\sqrt{a c}$. I want to define ...
2
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1answer
146 views

Solve Regularized Least Squares problems using Matlab optimization toolbox

I am trying to solve a least squares problem where the objective function has a least squares term along with L1 and L2 norm regularization. I am unable to find which matlab function provides the ...
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1answer
93 views

Checking the convexity or the concavity of a function

I have an optimization problem with the following objective function. $\max_{a^{l}_{n,k} b^l_{n,k}} \sum_{n=1}^{\overline{N_l}} b_{k,n} \frac{C_1}{C_2} \log_2 \bigg(1 +\frac{a_{k,n} h_{k,n}} ...
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2answers
744 views

Tikhonov regularization in the non-negative least square - NNLS (python:scipy)

I am working on a project that I need to add a regularization into the NNLS algorithm. Is there a way to add the Tikhonov regularization into the NNLS implementation of scipy [1]? [2] talks about it, ...
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0answers
47 views

Minimizing a quadratic form

I would like to minimize the following quadratic form: $$ f(\mathbf{\theta}) = (\mathbf{y} - \mathbf{\mu}(\mathbf{\theta}))^T \mathbf{\Sigma}({\mathbf{\theta}})^{-1} (\mathbf{y} - ...
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2answers
86 views

Nonconvex Optimization

Consider the following optimization problem: $\text{max}_{p} \quad ||p||^2 \\ s.t: x\geq 0\\ p\in D$ where $D$ is a convex set. Is this problem $\mathcal{NP}$-hard?
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1answer
48 views

Optimal linear transform in a simplex

I'm looking for a simple method to find a linear transform that minimizes $$ \text{argmin}_T F(T): T \in \mathbb{R}^{m \times n} ,\ T \ge 0 ,\ T \mathbb{1} = c \mathbb{1} ,\ \mathbb{1}^T T = ...
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2answers
66 views

minimize function with convex constraints

I need to solve the following problem: For a given p=(x0,y0,z0,w0) and arbitrary T. For example , let p=(0.8,0.1,0.06,0.04) and T=-1.2. I need to find a vector q=(x,y,z,w) with the minimum ...
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208 views

Computation Effort of Algorithms

Consider the strictly convex unconstrained optimization problem $\mathcal{O} := \min_{x \in \mathbb{R}^n} f(x).$ Let $x_\text{opt}$ denote its unique minima and $x_0$ be a given initial approximation ...
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1answer
294 views

minimization of a convex piecewise linear function [closed]

Let $$f(x) = \left\{ \begin{array}{l} {a_1}x + {b_1} & if\,0 \le {x_1} \le x \le {x_2}\\ {a_2}x + {b_2} & if\,{x_2} < x \le {x_3}\\ \vdots \\ {a_n}x + {b_n} & if\,{x_{n - 1}} < x ...
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0answers
56 views

Issues with CVX package for optimization

I am trying to use the cvx package for optimization. However, I am having some issues with it. I have a variable X which is a matrix but I cannot add $X^{-1}$ in the objective function. What should I ...
7
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2answers
265 views

How to deal with norm inequality constraints

I want to solve the (convex) optimisation task: $max_{r,z}\quad r$ subject to the following two constraints $r\|x_i\| - x_i^Tz \leq 0 \qquad \forall i=1,\dots, N $ $\|z\| \leq 1$ $r\geq0$ $r$ is a ...
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0answers
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Reduction for NP-hardness [duplicate]

Consider the following optimization problem: \begin{align} \text{Min}_{i\neq j\neq s\neq t} |x_i x_j-x_sx_t|\\ s.t: Ax=b\\ x\geq 0; \end{align} This problem can be seen as an instance of non convex ...
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2answers
188 views

NP-Completeness

Consider an instance of non-convexoptimization problem: It seems that this problem is NP-complete. How can I find a suitable reduction for this?
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53 views

Proper Algorithm for Image Recovery from Compressed Measurements in Office Spaces

Can anyone please suggest a proper algorithm for image reconstruction from undersampled data for office spaces ? Basis Pursuit (BP) works for sparse images which would not be a correct assumption for ...
5
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1answer
109 views

How to transform such an SDP to standard form

I plans to use CSDP to solve the following semi-definite problem: $$\min_{B, \beta}\operatorname{trace}(CB) \\ \text{s.t.} \ \operatorname{trace}(AB)=1 \\ \beta\geqslant 0 \\ \begin{bmatrix} 1 & ...
4
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2answers
138 views

Optimization algorithm selection for 3 variable integer

I have a cost function: $f(x,y,z) \rightarrow \mathbb{R}$ it is very expensive to evaluate $x,y,z \in \mathbb{Z}$ 0 < x < 10 0 < y < 30 0 < z < 100 I thought it was convex, not ...
3
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2answers
219 views

Quadratic Programming: Quadprog

Given a real-rectangular matrix $S$ and inorder to solve this simple quadratic programming problem: Minimize $w'S'Sw = ||S w||^2$ over $w$ subject to $e^Tw = 1$ and $w \geq 0$ using a solver I ...
3
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1answer
392 views

Biconvex optimization problems

Consider minimization of a biconvex function over a biconvex set. Is the biconvex optimization problems polynomially solvable?
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3answers
224 views

How to prove that a function is convex?

I'm calculating the gradient of a function with a symbolic math library called theano. Then I'm using gradient descent to find the minimum of the function. I'd like to prove that the minimum is a ...
0
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2answers
82 views

Analytical form of the minimum of a function with absolute values

I would like to find the analytical form of the point which minimizes the following function: $$ f(x_T) = \frac{1}{T} a_1 (x_T-x_0)^2 + a_2 |x_T-x_0| + T a_3 + \sum_{i=1}^M p_i \left[b_{1i} (x_T - ...
3
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2answers
278 views

Finding A and X such that AX = 0, X is positive non-zero, and A is sparse

I apologize if this is a naive question. I'm trying to create some boostrap data for a system of linear, ordinary differential equations at steady state. Since the equations represent the ...
7
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2answers
113 views

Why is the Dual problem preferred for Maximal Margin Classification?

The primal problem is $$\min_{w,b}\frac{1}{2}w^Tw$$ $$s.t. f_i(w)=1-y_i(w\cdot x_i+b)\leq0$$ Where $y_i=\pm1$. Instead of using Gradient Descent directly, the dual is usually solved: ...
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1answer
87 views

Confusion related to convexity of 0-1 loss function

I am a bit confused why the 0-1 loss function is not convex. What's wrong with it?
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5answers
963 views

MATLAB's CVX Package to minimize $\mathrm{trace}(S)+\mathrm{trace}(S^{-2})$

I would like to minimize in matlab the function $f(S)=\mathrm{trace}(S)+\mathrm{trace}(S^{-2})$ where $S\in \mathcal{M}_{m,m}$ symmetric positive definite which is definitely convex function. So I ...
1
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1answer
54 views

Confusion related to interior point method for optimization

I have this little confusion related to interior point method. In this method we use the log barrier function to approximate the real barrier which is not differential Now when we find the optimal ...
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1answer
55 views

Confusion while proving logdet function to be concave

I was going through this lecture related to convex optimization. It was proved that logdet function is concave. However, I didn't get the derivation at a part I didn't get how the step marked in ...
6
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1answer
160 views

Confusion related to convexity and concavity of a function

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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1answer
146 views

Robust Counterpart of an uncertain LP

Consider the following robust optimization problem: min c'x s.t.: $Ax\geq b \;\;\forall (A,b)\in \mathcal{U}$. Why can the robust counterpart of the problem be written in this form? $min_x{\{ ...
0
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1answer
30 views

Polynomial solvability [duplicate]

Consider the following optimization problem: Min$_{x}$ $\qquad \sum_{(i,j,t,s)\in I_r}||x_ix_j-x_tx_s||^2$ S.t.: $\qquad x\in \mathcal{C} ;$ where $x=(x_1,x_2,...x_n)$ and $\quad x_j\geq 0\;\; ...
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1answer
98 views

Feasibility checking

Consider the following optimization problem: $Min\;\;\; CX$ $AX\geq b$ $x_ix_j= x_s x_t\;\;\; i\neq j \neq s\neq t$ $x_j\geq 0;$ Where $A$ is the adjacency matrix and $C$ is a constant vector. ...
2
votes
2answers
112 views

polynomial time solvability

Consider the following optimization problem: $Min \qquad C^TX$ S.t.: $\qquad AX=0;$ $x_ix_j=x_kx_t$$\quad $for some $i\neq j\neq k\neq t$ $X=(x_1,x_2,...x_n)$ and $\quad x_j\geq 0\;\; ...
2
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1answer
71 views

Nonlinear bad constraints

I have an optimization problem with linear objective function. The constraints are in two different groups. The first set of constraints are linear while the second set is nonlinear. The nonlinear ...
4
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1answer
159 views

Numerical Methods for minimizing a Non-Differentiable Convex Function of Several Variables

I have a multi-variable convex continuous function which is not differentiable. I am interested to know about different numerical techniques, possibly also references to them, used for this. Read ...
2
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1answer
115 views

Extreme points from constraint expression of convex space

I'm looking for the extreme points of the convex set $S\subset [-1,1]^{n\times 3}$ with $r\in S$ such that \begin{equation} r_{i} \ge r_{k} \iff i\ge k, \end{equation} where the first inequality ...