Questions tagged [convex-optimization]

Convex optimization, 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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how to implement the correlation coefficient in cvxpy and how to rewrite cp.diff(cp.cumsum(x)/(cp.cumsum(x)[-1])

suppose that I have 2 sets of data, x and y,where x can be ...
Allen Zhang's user avatar
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Converting a Linear Program with TU Constraint Matrix to a Nonlinear Convex Model: Solver Performance?

I'm currently working on a large Mixed Integer Program (MIP) where the constraint matrix is Totally Unimodular (TU), allowing me to model it as a Linear Program (LP) for efficiency, as total ...
graphtheory123's user avatar
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CVXPY ECOS and SCS solver stuck in local optima for convex problem?

Consider the following optimization problem: \begin{equation*} \begin{aligned} & \underset{z_{k, t}}{\text{maximize}} & & \sum_{k} \sum_{t} \text{concave-function}(z_{k,t}) \\ & \text{...
rutkov's user avatar
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Global optimizers handling minimization of an expression arising from the likelihood of a multivariate normal

I am interested in converting the following optimisation problem into a form that an exponential cone and/or SDP solver such as MOSEK can handle. This is a multivariate version of the question I ...
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Can ADMM be applied to "latently coupled" variables?

I've been studying a paper where the authors employ the ADMM in a way that has left me somewhat perplexed. The paper focuses on addressing a robust principal component analysis (RPCA) problem, ...
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Differences between non-convex and convex optimization problem with l0-Norm Regulization

I'm currently in the process of writing my bachelor's thesis and trying to deal with the theory behind the model in this paper Risk-calibrated Super-sparse Linear Integer Model (Berk Ustun and Cynthia ...
user13121's user avatar
5 votes
2 answers
167 views

Global optimizers handling minimization of expressions like $\log{v}+\frac{1}{v}$

Consider the simple problem of maximum likelihood estimation of the variance of a mean zero normal distribution. The expression to be minimised is: $$N \log{v}+\frac{1}{v}\sum_{n=1}^N{b_n^2},$$ where $...
cfp's user avatar
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Is it possible to show that this problem is convex?

$\DeclareMathOperator{\Tr}{Tr}\DeclareMathOperator*{\argmax}{\arg\!\max}$Consider the following problem $$ \begin{aligned} \min_x \quad & \Tr(WF(x)) \\ \textrm{s.t.} \quad & 0 < x \leq 1 \...
mhdadk's user avatar
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Which combinations of python modeling libraries and solvers support logarithmic objective functions?

As I understand it, if a modeling library (pyomo, amplpy, cvxpy, etc.) allows one to define a logarithmic objective function a solver may not support it. And even if a solver does support a ...
Nick Laws's user avatar
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How is semidefinite programming a special case of convex programming?

In this image from Wikipedia, semidefinite programming is presented as a special case of convex programming. I do not see how this can be. Consider the following two constraints (where $\succeq$ means ...
Erel Segal-Halevi's user avatar
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Lagrange multiplier associated to an active inequality constraint

Why is the Lagrange multiplier associated to an active inequality constraint is positive. How can we see this from the KKT conditions?
DSPinfinity's user avatar
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How is this problem quasi-convex?

I'm currently reading the following paper B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M. I. Jordan, and S. S. Sastry, “Kalman filtering with intermittent observations,” IEEE Transactions ...
mhdadk's user avatar
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3 answers
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Do convex quadratic problems always have sparse solutions?

It is known that a feasible bounded linear program with $m$ constraints always has a solution with at most $m$ non-zero variables (a basic feasible solution). Since the number of constraints might be ...
Erel Segal-Halevi's user avatar
2 votes
2 answers
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Can the following problem be solved recursively?

Consider the following problem \begin{equation} \begin{aligned} \min_{x,y,z} \quad & \sum_{i=0}^1 \sum_{j=0}^1 \sum_{k=0}^1 a_{ijk} \cdot f_{ijk}(x,y,z), \\ \textrm{s.t.} \quad &...
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Convexity of p power of the q norm (0<p<1, q>1)

I encountered a minimization problem involving the following function: $f(\mathbf{x})=\|\mathbf{x}\|_q^p$ Here, $q>1$ and $0<p<1$. Naturally, each entry of $\mathbf{x}$ is greater than $0$. I ...
Piko Mone's user avatar
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Bilinear programming

Let us assume I have an optimization problem as follows: The first feasible region is an ellipsoid, meaning that a and b belong to a known ellipsoid. The second feasible region is polyhedral (a set ...
naghi pakdaman's user avatar
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Solving a max-min convex optimization problem with interior-point methods

I would like to solve the following problem: \begin{align} \text{ minimize } && t \\ \text{ subject to } && f_i(x) - t \leq 0 \text{ for all $i\in 1,\ldots,n$,} \\ && 0\leq ...
Erel Segal-Halevi's user avatar
2 votes
1 answer
97 views

Basic question about definition of a convex optimization problem

In lecture notes by Nemirovsky and BenTal (2023), I found the following definition of a convex optimization problem: MY QUESTION: why do we need both the functions $f_i$ and the domain $G$? ...
Erel Segal-Halevi's user avatar
2 votes
2 answers
113 views

How can I relax the equality constraint in this problem?

Consider the following problem \begin{equation} \begin{aligned} \min_{x,y} \quad & f(x,y), \\ \textrm{s.t.} \quad & \exp(x) + \exp(y) = 1 \end{aligned} \tag{1} \end{...
mhdadk's user avatar
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transform minimize weighted sum of absolute value into a linear optimization

For example, we have an optimization problem $$ \min \sum_{i=1}^{n} |w_{i} - a_{i}| b_{i} \quad \text{s.t.} \quad \sum_{i=1}^{n} c_i w_i = 0 $$ and $a_i, b_i, c_i$ are given. How to convert it into a ...
Pique's user avatar
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Convex programming without a lower bound on the feasible region size

I need to solve a convex minimization problem of the form: minimize $f(x)$ such that $g_i(x)\leq 0$, where the $g_i$ are convex functions given by a value oracle. As far as I know, to use the ...
Erel Segal-Halevi's user avatar
0 votes
1 answer
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Convex approximation of an expression with fraction for CVX

I have the optimization problem $$\underset{\mathbf{x} \in \Bbb C^N}{\max} \left| \frac{\mathbf{x}a-b}{\mathbf{x}c+b} \right|^2$$ where $a$, $b$ and $c$ are some scalars. I want to solve this ...
Muhammad's user avatar
1 vote
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141 views

MOSEK via fusion vs API vs CVXPY

In Python, I would like to solve a collection of problems, that are all solvable via MOSEK's conic optimization solvers (ExpCone, SOCP, etc.) I have tried CVXPY. I get very robust and reliable results,...
independentvariable's user avatar
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1 answer
263 views

Convex approximation of an expression

I am trying to transform an expression given by $$ \operatorname{trace} \left( {\bf{X} } \right) + \left( \sum_{n=1}^N \mathcal{R}(x_n) \right) $$ into convex from where $\mathbf{x}$ is complex in ...
Muhammad's user avatar
1 vote
0 answers
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Distributionally Robust Stochastic Programming - Help with derivation

I've been working through this book on robust optimization of electric energy systems, and in particular chapter 4 on distributionally robust optimization. In following the derivation of section 4.2.1....
asfiwefewrno's user avatar
5 votes
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523 views

How to write this objective in CVXPY for quasiconvex programming?

I have the following objective that I want to maximize: \begin{equation} \max_{U_T\in \mathbb{R}, x\in\mathbb{R}^T} J_\alpha(U_T) = \frac{\alpha}{\alpha-1}\log\left(\frac{\cosh(U_T)}{\cosh(\alpha U_T)^...
Uomond's user avatar
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I need help with finding a library or tutorial to parallel optimize given a black box function, assuming it's convex

I have a black box function to optimize with respect to 1-D input,$\beta$, and I also have other inputs, which I don't need to optimize, say (x,y). So, I need to optimize f($\beta$, x,y) over $\beta$. ...
Thiha Aung's user avatar
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log-log regression as reward function in optimization problem

Consider the model $\hat{y}_t = e^{\text{trend} + \text{seasonality}} \prod_k^K x_{k, t}^{b_k}$ where $K$ denotes different investment alternatives. You can think that trend and seasonality are ...
pete lewis's user avatar
2 votes
1 answer
223 views

Quadratic optimisation with $\ell_1$ constraints with CVXPY

Crossposted on Mathematics SE I seek to minimize a convex quadratic objective subject to linear and $\ell_1$-based equality constraints. When I turn to CVXPY, an error is raised indicating that it ...
jam123's user avatar
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1 vote
2 answers
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Relaxing non-affine equality constraints in convex optimization

Consider the convex function $f$. In section 4.2.1 in these lecture notes, the author writes: 4.2.1 Relaxing non-affine equality constraints For functions $g_i(x)$, $i \in \{1,\dots,d\}$ that are ...
mhdadk's user avatar
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What is the use of solvers that return approximately-feasible solutions?

Common methods for solving convex programs return solutions that are only approximately-feasible solutions. Here is an example (from lecture notes by Nemirovski on interior-point methods for convex ...
Erel Segal-Halevi's user avatar
3 votes
1 answer
60 views

Finding a starting ellipsoid and a minimum volume to approximate a convex optimization problem

Suppose we have a convex optimizatiom program: \begin{align} \min &\quad f_0(x)\\ s.t. &\quad h_i(x) = 0 && i=1,\ldots, p\\ &\quad g_i(x) \leq 0 && i=1,\ldots, m\\ &\...
eden hartman's user avatar
3 votes
1 answer
83 views

Is it possible to make a posynomial concave using a change of variables?

Note: this question was already posted on Math.SE but received no answers, so I'm re-posting it here for better reach. Consider the following posynomial with respect to the variables $x_1,\dots,x_n$: ...
mhdadk's user avatar
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0 votes
1 answer
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Algorithms for maximizing the sum of power functions with linear constraints?

I’m working on an optimization problem that arises from maximizing the return obtained from investing in different marketing levers. The return from each lever exhibits diminishing returns, and is ...
Carlos Zanini's user avatar
3 votes
0 answers
52 views

Are there algorithms for minimizing a sum of convex and non-convex/non-concave functions?

Consider the problem \begin{equation} \begin{aligned} \min_{x} \quad & f(x) + g(x), \\ \textrm{s.t.} \quad & x \in X \end{aligned} \tag{1} \end{equation} where $X \subset \...
mhdadk's user avatar
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Is a posynomial concave under the following conditions?

Update In case anyone is interested, this question has been answered here. Consider the following posynomial with respect to the variables $x_1,\dots,x_n$: \begin{align} f(x_1,\dots,x_n) &= \sum_{...
mhdadk's user avatar
  • 601
0 votes
1 answer
99 views

Linear Programming faster solver: CPLEX or Gurobi?

Which one is the best solver for solving large-sized problems in Linear Programming, CPLEX or Gurobi? Which one is faster?
Abbas Khademi's user avatar
2 votes
0 answers
68 views

Approximating an LP with an exponential number of variables and an almost-separation-oracle to its dual

Problem settings: we have $n$ agents and a set $\mathcal{S}$ of possible world-states, where the size of $\mathcal{S}$ is exponential with respect to $n$. Each agent $j$ has a utility function $u_j\...
eden hartman's user avatar
2 votes
1 answer
75 views

Question About Fritz John Theorem and Slater Constraint Qualification

Background Information I am studying constraint qualifications. Here are two theorems leading to my question: Theorem 1$\space\space\space\space$ [Fritz John Theorem] Suppose that $f, g_1, \dots, g_k$...
Beerus's user avatar
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3 votes
2 answers
175 views

When Biconvex function is Pseudoconvex function?

Is a Biconvex function f(x,y):=yg(x), (where g is a convex function, y>=0), Pseudoconvex function?
Abbas Khademi's user avatar
-1 votes
1 answer
68 views

How to linearize the multiplication of variables and transform this into an MILP?

Let $C=10$, $U=50$ $P_c,c=1,\cdots,C$ and $\alpha_{u,c},u=1,\cdots,U,c=1,\cdots,C$ are optimization variables $\alpha_{u,c}$ is binary $\sigma_{u,c}$, $d_{u,c}$ are known parameters $\min \sum_{c=1}^...
KGM's user avatar
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3 votes
1 answer
90 views

Dual norm definition: adding new constraints

For some $c >0$ and $z \in \mathbb{R}^n$, the optimal value of \begin{align} \begin{array}{cl} \sup_{x \in \mathbb{R}^n}& z^\top x \\\text{s.t.}& \lVert x \rVert \leq c \end{array} \end{...
independentvariable's user avatar
1 vote
0 answers
25 views

Nesterov Acceleration in Proximal Iteration with Strictly Positive Variable?

I have a cost function of the form $f(x) - \log(x)$, where $f$ is some smooth function mapping $\mathbb{R}^n_+\to\mathbb{R}$and $x$ is restricted to be nonnegative by the problem constraints (and for ...
John Madden's user avatar
2 votes
0 answers
81 views

Does the value function of a quadratic program stay convex when adding constraints?

I am interested in the value function of a quadratic program of the form $$ v(y)=\min_x \frac{1}{2} x^\top Q(y) x, $$ subject to a linear equality constraint $$ E(y)x=d(y), $$ and a linear inequality ...
user_lambda's user avatar
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0 answers
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Why is CVaR positively homogeneous?

I am reading the article "Some remarks on the Value at Risk and the Conditional Value at Risk" by G. Pflug. The author claims that CVaR is positively homogeneous. That is, Given a loss ...
pele's user avatar
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0 answers
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When the Weighted Sum method yields a unique solution in MOLP?

I would like to ask if there are some specific conditions under which the Weighted Sum method yields a unique solution for MOLP or more general for convex problems. Let us assume that the number of ...
Tassos's user avatar
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0 answers
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How do I implement this convex problem in CVXPY?

I am looking to implement the following optimization problem in CVXPY. $$ \max _{x_t} x_t' \mu - \frac{\gamma}{2} x'_t \Sigma x_t - x'_t\Lambda \Delta x_t $$ where $\Delta x_t := x_t - x_{t-1}$ and $\...
Lydia's user avatar
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1 vote
1 answer
118 views

Equivalent condition for indicator function

We have the following two conditions: $C1.$ $l(x) \geq 0$ for all $x\in \mathbb{R}^n$, $C2.$ $l(x) \geq 1$ for all $x\in \mathbb{R}^n$ such that $a+b^Tx \leq 0.$ Here $l(x) = \begin{bmatrix} x^T 1 \...
Cherryblossoms's user avatar
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0 answers
62 views

What the dual constraints mean?

I have a primal problem that is a network-flow: $$ min_x \sum_{a \in A} c_a x_a \\ \text{s.t.} \sum_{a \in A:n = v} X_{a}-\sum_{a \in A:n' = v} X_{a} = \begin{cases} \;\;\;1, \; v=n^{in} \\ -1, \; v=n^...
orpanter's user avatar
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2 votes
1 answer
66 views

epigraphs for quadratic constraints

I have a constraint of the following form \begin{equation} x^{\top}x + y^{\top}y \leq t \end{equation} where x, y are vector variables and t is a scalar variable. I can augment the variables x and y, ...
Kumar's user avatar
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