# 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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### 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 $G$? ...
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### 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{...
1 vote
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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 ...
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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 ...
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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 ...
1 vote
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### 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,...
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### Convex approximation of an expression

I am trying to transform an expression given by $$\operatorname{trace} \left( {\bf{X} } \right) + \left( \sum_{n=1}^N x_n \right)$$ into convex from where $\mathbf{x}$ is complex in nature.
1 vote
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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....
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### 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)^...
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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 $\beta$. ...
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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 ...
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### 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 ...
1 vote
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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 ...
1 vote
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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 ...
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### 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\\ &\...
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### 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 $x_1,\dots,x_n$: ...
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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 ...
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### 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 $f, g_1, \dots, g_k$...
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### 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?
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1 vote
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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 \... 0 votes 0 answers 56 views ### What the dual constraints mean? I have a primal problem that is a network-flow: $$min_x \sum_{a \in A} c_a x_a \\ \text{s.t.} \sum_{a \in A:n = v} X_{a}-\sum_{a \in A:n' = v} X_{a} = \begin{cases} \;\;\;1, \; v=n^{in} \\ -1, \; v=n^... 2 votes 1 answer 59 views ### epigraphs for quadratic constraints I have a constraint of the following form \begin{equation} x^{\top}x + y^{\top}y \leq t \end{equation} where x, y are vector variables and t is a scalar variable. I can augment the variables x and y, ... 0 votes 1 answer 59 views ### Min-convex function as constraint I have a constraint that is as follows:$$ Ax - f(x) \leq 0 $$where f(x)=min_y(g(x,y)). Which is convex. I can even get the gradient in x. How can I reformulate my constraint? or what ... 3 votes 2 answers 153 views ### Simplest Quadratic Programming algorithm for teaching Can anyone recommend a straightforward quadratic programming (QP) algorithm suitable for an undergraduate engineering class? I'm interested in finding an algorithm that they can easily grasp and ... 1 vote 1 answer 113 views ### Rational LP, its Rational solution and a minimum precision Suppose we have an LP with rational coefficients. To my knowledge, this implies that the optimal solution to that LP is also rational. In other words, every variable may be written as:$$x_{i}^{\star} ... 2 votes 1 answer 45 views ### When is$\max_x\{f(g(x))\} = f(\max_x\{g(x)\})$? What is the requirements on$f$and$g$in order for$\max_x\{f(g(x))\} = f(\max_x\{g(x)\})$to be correct? Equivalently, when is$\min_x\{f(g(x))\} = f(\min_x\{g(x)\})$? Any reference for describing ... 1 vote 1 answer 55 views ### Quadratic optimization with non-constant coefficients I have a series of functions (very similar to convex quadratic equations, see the first comment below)$f_1(x), f_2(x), \dots, f_n(x)$. Each of these functions touches the$x$-axis at$a_i$, which can ... 1 vote 2 answers 60 views ### Solving regularized least squares problem using black-box computation of$\mathbf{A}\mathbf{x}$and$\mathbf{A}^T\mathbf{x}$Let$\mathbf{A} \in \mathbb{R}^{n \times n}$. I'm working in a problem where I have a black-box algorithmic solution to compute the products$\mathbf{A}\mathbf{x}$and$\mathbf{A}^T \mathbf{x}$given ... 2 votes 2 answers 126 views ### ADMM diverges on L1 regression TLDR: Why does ADMM diverge when solving$\ell_1$regression? Introduction I am learning about convex optimisation and wanted to solve a simple exercise that I am having issues with. I want to solve a ... 1 vote 1 answer 93 views ### Sensitivity analysis for decision vectors in convex programming Can we perform sensitivity analysis on the decision variables for the perturbed right-hand side of the constraints in a convex/nonlinear program? I know a basic result regarding the sensitivity of the ... 0 votes 0 answers 40 views ### Convex relaxation of an inner product for optimization I am trying to solve the following problem. The optimization variables are$x \in R^{+,I}$,$y \in {R}^{+,I x S}$and$z \in {R}^{+,J x S}$.$a_{i}, b_{j}, c_{i}$and$d_{ij}$are parameters and all ... -1 votes 2 answers 133 views ### Reformulate this constraint optimization problem such that I do not have to divide 2 variables? I have a constraint optimization problem as follows: I need to assign$m$tasks to$n$days, with$n \geq m$. Each day can host 0 to$m$tasks. Each task either belongs to type$A$or$B$. I want to ... 3 votes 1 answer 204 views ### Solver for quadratically constrained mixed-integer linear programs I have an optimization problem with vectors$x$,$y$, and$z$, where$x$is an integer vector. My objective function is linear (i.e.$\|y\|_1$), but one of my constraints is quadratic ($x^Ty \leq z$). ... 2 votes 0 answers 180 views ### Distributed optimization with coupled inequality constraints Consider the optimization problem: \begin{equation} \begin{array}{l} \min_{(x,y)\in \mathbb{R}^2_{+}} \quad x_{1}a_{1} + x_2a_{2} \\ \text {subject to } \quad\; y_{2} \ge \frac{1}{{x}_2}, \\ \quad \... 4 votes 0 answers 85 views ### The study of directional derivatives for functions that are minimums of convex functions Has there been any research on the topic of directional derivatives of functions that are minimums of convex functions? 2 votes 1 answer 106 views ### Separation oracle gives optimisation Suppose I want to minimize a linear function$f$over a convex set$K$and I have only access to a separation oracle, that is, given a point, the oracle returns yes if the point is in$K$and ... 2 votes 1 answer 64 views ### Does this kind of "partition" have a name? Consider a convex polyhedron$A$. Assume we have subsets$A_1,\ldots,A_n$of$A$that are themselves covex polyhedra and are mutually disjoint except maybe sharing an edge, and that their union gives$...
My question is related to a previous one: Dedicated solver for convex problems To minimize a convex function of the form $~f(x_i) = \left[ C + mx_i + \frac{s}{x_i+t} \right]^p$ with various ...