Questions tagged [convex-optimization]

Convex minimization, a subfield of optimization, studies the problem of minimizing convex functions over convex sets. The convexity property can make optimization in some sense "easier" than the general case - for example, any local minimum must be a global minimum.

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1answer
68 views

How can I linearise this nonlinear proportional relation constraint?

I have an optimisation problem as follows... \begin{equation} \begin{array}{*{35}{l}} \underset{d_{u,c}}{\max}\hspace{1mm}\hspace{1mm}\sum_{u=1}^{U}\sum_{c=1}^{C}d_{u,c}\omega_{u,c}\\ \text{}\text{...
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1answer
103 views

How to prove that the second-stage value function is convex?

I am wondering to know how it is possible to prove that the second-stage value function in a two-stage stochastic program is convex on $x$ and $\xi$? A two-stage stochastic program can be defined as \...
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2answers
118 views

Separating hyperplanes for a convex cone

Let $W$ be a fixed matrix. Define $$\operatorname{pos}W \triangleq \{t \mid Wy =t , y≥ 0\}.$$ It is called the positive hull of $W$. It represents the set of right-hand sides that can be obtained by a ...
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2answers
98 views

Should I process the data or add a new constraint to achieve the target?

I have an MILP as below $\begin{equation} \begin{array}{*{35}{l}} \underset{d_{u,c}}{\max}\hspace{1mm}\hspace{1mm}\sum_{u=1}^{U}\sum_{c=1}^{C}d_{u,c}\omega_{u,c}\\ \text{}\text{subject to }\text{ C1:}...
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1answer
83 views

Recommended python solver for an online optimization problem

I need to implement a load scheduling algorithm that involves solving an online optimisation problem from a research paper for my Real time systems course. This convex optimisation problem is setup ...
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0answers
64 views

Decomposition of Polyhedra

There is no doubt that clear examples consolidate the understanding of concepts being learnt. I am new to finding the structure and decomposition of a polyhedra. Suppose that we have the system $$ \...
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2answers
147 views

Piecewise linear and global optimization

I am new to OR, and apologies if my mathematical notation is not clear. I have tried my best to keep it concise and given an explanation with numerical data. I would like to understand: Can this ...
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1answer
79 views

Following code doesn't work in matlab with CVX

Given the following problem \begin{align}\min&\quad x_1+2x_2+3x_3+4x_4+\sum_{i=1}^4x_i\ln(x_i)\\\text{s.t.}&\quad e^\top x=1\\&\quad x\geq0\end{align} I was asked to solved the dual ...
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4answers
339 views

Disciplined convex programming representation of $x\sqrt{1-x}$

Anyone have an idea for a disciplined convex programming (DCP) representation of the concave function $x\sqrt{1-x}$, which is defined over the domain $[0,1]$? The Taylor series about $x=0$ is $$x - \...
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1answer
49 views

Find a dual problem with one dual decision variable to the problem of finding the orthogonal projection of a given vector

Given the set $T_{\alpha}=\{x\in\mathbb{R}^n:\sum x_i=1,0\leq x_i\leq \alpha\}$ For which $\alpha$ the set is non-empty? Find a dual problem with one dual decision variable to the problem of finding ...
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1answer
93 views

Find the dual problem of $\min_x\{||x-a_1||+||x-a_2||+||x-a_3||,a_i\in\mathbb{R}^n\}$

Find the dual problem of $$\min_x\{||x-a_1||+||x-a_2||+||x-a_3||,a_i\in\mathbb{R}^n\}$$ I've tried the following but got stuck $$\min_x\{||x-a_1||+||x-a_2||+||x-a_3||,a_i\in\mathbb{R}^n\}=\min_{x,z_i}...
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51 views

Prove $\sum_{i=1}^{m}\lambda_i^*\leq\frac{f(\hat{x})-f^*}{\underset{i=1,\ldots,m}{\min}(-g_i(\hat{x}))}$

Consider the primal problem \begin{align}f^*=\min&\quad f(x)\\\text{s.t.}&\quad g_i(x)\le0\tag P\end{align} where $f,g_i$ are convex functions. Suppose there exists $\hat{x}$ such that $g_i(\...
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1answer
80 views

Prove that $x^*$ is an optimal solution where $f_0$ is $C^1$ and convex and $f_i$ are $C^1$ and strictly convex functions

Let $x^*$ be a feasible solution of the following convex optimization problem \begin{align}\min&\quad f_0(x)\\\text{s.t.}&\quad f_i(x)\leq0,i=1,\ldots,m\end{align} where $f_0$ is $C^1$ and ...
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91 views

Prove Non-Homogeneous Farkas' Lemma

Let $A\in\mathbb{R}^{m \times n}, c\in\mathbb{R}^{n}, b\in\mathbb{R}^{m}, d\in\mathbb{R}$. Suppose that there exists $y\geq0$ such that $A^Ty=c$. Question: prove that exactly one of the following is ...
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0answers
38 views

active set method guaranteed convergence

I'm using Active Set Method to solve a nonlinear optimization function minimizing a convex function over a polyhedron of 2 linear inequalities starting at an interior point $x_o$ At this point is it ...
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2answers
94 views

Let $A\in\mathbb{R}^{m\times n},c\in\mathbb{R}^n$. Show that exactly one of the following two systems is feasible:

Let $A\in\mathbb{R}^{m\times n},c\in\mathbb{R}^n$. Show that exactly one of the following two systems is feasible: $Ax\geq0,x\geq0,c^Tx>0$ $A^Ty\geq c,y\leq0$ Assume that A is feasible meaning $...
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0answers
69 views

How to linearize this multiplicative constraint?

I have a constraint in the form $\sqrt{|\sum_{c\in C}{h_cw_c}|^2}\ge\sqrt{x}\zeta$ Here, $h_c$ is s row vector (know), $w_c$ is a column vector (variable). $x$ and $\zeta$ are also optimization ...
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1answer
35 views

How to convert an element of a variable to a convex constraint using binary variables?

I defined a complex variable in cvx, but I want to restrict the first element of the variable to be larger than the max of the variable, but it doesn't work. Someone told me to transform it using a ...
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1answer
60 views

How can I model this Hyperbolic constraint?

In this problem, $\beta_u$, $w_{u,c}$ (a vector of complex elements), $x_u$ are optimization variables. Now, $||2\sqrt{\frac{\pi_u}{2}}\beta_u; h_{u,c}^{\rm H}w_{u,c}-\frac{1}{2\pi_u}x_u-1||_2\le h_{u,...
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1answer
67 views

How to model these constraints correctly

$W$ is a vector of $N$ complex elements. $D$ is a binary variable The requirements are: when $D==1$, $L_{\min}\le ||W||_2^2\le L_{\max}$ and when $D==0$, $||W||_2^2=0$ I have formulated the following ...
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3answers
332 views

Find the farthest point in hypercube to an exterior point

Let $\mathcal{U} = \{ [x_1, ..., x_n] \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$ be the unit hypercube and $C \in \mathbb{R}^n\setminus\mathcal{U}$ fixed. Let us consider the following problem $$ \max_{X ...
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0answers
24 views

Stationary conditions for intersection

I wondered about this question for sometime. Definition of Stationarity (P) $\mbox{min} f(x)$ s.t $x\in C$ Let $f$ be $C^1$ function over a closed and convex set $C$ . then $x^*$ is called a ...
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1answer
84 views

How to prove this convex-optimization problem?

I am struggling with the following optimization problems. Problem 1 \begin{align}\max_{\alpha, s_1, s_2}&\quad s_1 + s_2 - \gamma (s_1 (K_1 +c_1 + s_1) + s_2 (K_2+ c_2 + s_2) + 2\alpha K) +C\\\...
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2answers
207 views

How to solve this convex problem heuristically?

I have the following problem $$\max_{X_{i,j},i\in N_{U},j\in N_{B}}\sum_{i=1}^{N_U}\sum_{j=1}^{N_B}R_{i,j}X_{i,j}$$ $$\text{subject to}$$ $$a_{\min}\le\sum_{j=1}^{N_B}X_{i,j}\le a_{\max}, \forall i$$ $...
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Is this a valid strong polynomial algorithm for deciding LP feasibility?

Let $$A \cdot X + B \preceq 0$$ be a system of linear inequalities with $X \in \mathbb{R}^n$ $A\in \mathbb{R}^{m\times n}$ and $B \in \mathbb{R}^m$ where $m \geq n$. According to Farkas lemma, exactly ...
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1answer
131 views

Can we get the closed-form solution for this problem?

Can we get the closed-form solution for this problem? \begin{align} \min&\quad\sum_{i=1}^N\frac{K_i}{x_i\log_2(1+\frac{Q_i}{x_i})}\\ {\rm{s.t.}}&\quad\sum_{i=1}^N x_i\le X \end{align} wherein $...
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0answers
71 views

Polyhedra to Simplex by using convex combination of vertices

Optimization problems over linear constraints (defining a convex polyhedron) can be written as optimization over a simplex in a higher dimension. Let $\mathcal{P}$ be a bounded polyhedron, and the ...
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33 views

Optimization of strongly convex functions with approximate evaluations of gradient and Hessian

Suppose I want to find the minimum of a differentiable, strongly convex function $f:\mathbb{R}^n\to\mathbb{R}$ with constant $\mu>0$. That is, for all $x,y\in\mathbb{R}^n$, I have that: $$f(y) \geq ...
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62 views

Optimizing with a logistic function

I have a system in which I want to maximize the value of some function $f(x_T, y_T)$. The time evolution of the system is described by some functions: $$ \begin{align} \frac{dx}{dt}&=\alpha \frac{...
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1answer
61 views

How to evaluate the convexity of an optimal control problem?

Can we consider an optimal control problem, a convex optimization problem like static optimization problems? If it is possible, under what conditions, will this problem be a convex problem? For ...
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0answers
55 views

How to solve this problem by Lagrange duality?

This is a convex problem and although it can be well solved by CVX, I want to know how it can be solved by the Lagrange duality method. The derivations with regard to $L_k$ and $B_k$ are constants, ...
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44 views

$\nabla_y\nabla_vf(x^*)\geq0$ for any concave $f$ if and only if $y=-v$

$f:\mathbb R^3\to\mathbb R$ is an arbitrary concave function. $H$ is a plane. $v$ is a given vector on $H$. $x^*=\max_{x\in H} f(x)$ Prove that $\nabla_y\nabla_vf(x^*)\geq 0$ if and only if $y=-v$. I ...
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1answer
63 views

Quasi-convex function must be “partially monotonic”?

$f(x)$ is quasi-convex, $$x^*\in\arg\min_{x\in C}f(x).$$ How to prove that, for any $a\in C$, $f(x) $ is weakly monotonic in the direction of $(x^*-a)$? Is this simple result a part of an ancient ...
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33 views

Linear functions in Lenstra's algorithm

I had asked this question at MathOverflow and was pointed here. I'm working on implementing Lenstra's algorithm. At the bottom of p.5 (at "construct $n+1$ linear functions"), he says to ...
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1answer
129 views

Convexity of the variance of a mixture distribution

$X$ is a random variable that is sampled from the mixture of uniform distributions. In other words: $$X \sim \sum_{i=1}^N w_i \cdot \mathbb{U}(x_i, x_{i+1}),$$ where $\mathbb{U}(x_i, x_{i+1})$ denotes ...
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741 views

Difference between exploration and exploitation in Simulated Annealing algorithm

In evolutionary algorithms, two main abilities maintained which are Exploration and Exploitation. In Exploration the algorithm searching for new solutions in new regions, while Exploitation means ...
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2answers
194 views

Can we use reinforcement learning and convex optimization to solve an optimization problem?

For an optimization problem, there are multiple-type variables should be optimized. Can we use the convex optimization method to solve a subproblem of partial variables, and then, with the obtained ...
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1answer
87 views

Conditions required for strong duality to hold for SDPs

According to Wikipedia, strong duality holds when "the primal optimal objective and the dual optimal objective are equal." What are the necessary conditions for strong duality to hold in ...
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1answer
86 views

Relationship between extreme points and optimal solutions of SDPs

Consider this to be our SDP problem: Minimize $\langle C, X \rangle$ such that $\langle A_i, X \rangle \ge b_i$ for all $i \in [m]$ and $X \succcurlyeq 0$. For SDPs, is there a relationship between ...
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0answers
65 views

Can every convex problem use Lagrangian dual method?

If not all constraints satisfy equalities, does Lagrangian dual method make sense to a convex problem?
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1answer
104 views

Is a convex or MILP (without big-M) formulation possible for this problem

Assume we are given a directed acyclic graph (DAG) $G(V, A)$, where $|V| = n, |A| = m$, and the graph contains a source node $\mathbf{s}$ (i.e. every node in $V \backslash \mathbf{s}$ is connected by ...
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0answers
80 views

Question on quadratically constrained quadratic program

If the constrained optimization problem is a quadratically constrained quadratic program of the form \begin{align}\min&\quad x^HQx-a(x+x^H)+b|z^Hx|^2\\\text{s.t.}&\quad\|x\|^2\le1\end{align} ...
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2answers
129 views

DCP representation of a convex quotient of affine functions

I am trying to represent the following inequality: $$\frac{x}{1-x} \leq y \qquad\mathrm{with}\qquad 0<x<1$$ The function on the left is convex (its second derivative is always positive over ...
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140 views

Find a point inside non-empty difference of ellipsoids

Given two ellipsoids \begin{align}\mathcal{E}_1 &= \{ X \mid X^\top A_1 X + 2B_1^\top X + C_1 \leq 0\}\\\mathcal{E}_2 &= \{ X \mid X^\top A_2 X + 2 B_2^\top X + C_2 \leq 0\}\end{align} are ...
4
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1answer
60 views

How to check for convexity of the inequality constraint $−x^2+y−1\ge0$ for a minimization objective function?

I checked the Hessian which is $\begin{bmatrix}-2&0\\0&0\end{bmatrix}$ which is negative semidefinite but according to the sketch of the function it is convex. What am I missing?
3
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1answer
166 views

Approximation methods for a mixed integer convex optimization problem

I have a convex objective function, e.g., minimizing the negative entropy function. My constraints are also linear. The only issue is that I also have binary variables. I am currently aware of AIMMS'...
5
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1answer
114 views

Which solver solves PSD constrained convex non-linear problem

I have a problem with a vector variable $w \in \mathbb{R}^n$ and a symmetric matrix variable $V \in \mathbb{R^{n \times n}}$. I am solving a problem which is roughly like: \begin{align} \begin{array}{...
4
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1answer
105 views

Minimize a convex function over a sphere

Problem description Let $\mathcal{C} = \{X \in \mathbb{R}^n \mid g(X) \leq 0\}$ with $g(X)$ a convex function. Suppose I need to solve the feasibility problem, for a given $r>0$ $$ \exists ^?X \...
3
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0answers
71 views

Strong Duality and Slater Condition

I am studying the Duality Chapter of Convex Optimization by Boyd. Is it possible that strong duality holds for non-convex optimization? If yes, is there any specific condition? And, what is the ...
4
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1answer
151 views

Cutting-planes application procedure for a specific problem

Sort of following up with this question. I reformulated another model to make it convex and possibly solve it with some cut generation method. I would like to double-check whether I am doing it ...