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Questions tagged [reformulation-linearization]

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Converting a function composing of multipe pieces into a linear equation

I have a variable (alpha) which depends on some other binary variables, denoted as X_i. So, for some combination of other variables, alpha may take a value (Beta_j). I added some auxillary variables (...
Sam's user avatar
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0 answers
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Removing min operator from max objective

I am reading a paper where the authors made the following statement on page 16: I wonder why dropping the 'min' operator inside a 'max' objective function is a valid operation here. Any explanation ...
Apurba Saha's user avatar
2 votes
1 answer
244 views

Linearize piecewise function without big-M constraints

I have been attempting to solve a maximization problem where there is a piecewise function in the objective. Something like: $\sum_{n}(1-prob_{n})(1+x_n)$ Where $prob_{n} = $ \begin{cases} 0.25,...
akkha's user avatar
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2 votes
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Addressing Variable Multiplication in Constrained Infinity-Norm Maximization with Hypercube & Polyhedron Constraints

I am reaching out to this knowledgeable community for assistance with a complex optimization problem that I have been investigating. Here is the formulation of the problem I'm addressing: $$\tag{1} \...
Diego Fonseca's user avatar
1 vote
0 answers
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Convex quadratic maximization over cartesian product of simplices

Suppose we are maximizing $f(x^1,\ldots,x^t)= \begin{bmatrix}{x^1}^\top & \ldots & {x^t}^\top \end{bmatrix}^\top Q \begin{bmatrix}{x^1}^\top & \ldots & {x^t}^\top \end{bmatrix}$ ...
independentvariable's user avatar
0 votes
1 answer
72 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 ...
orpanter's user avatar
  • 517
2 votes
1 answer
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Automatic Reformulation Tools For AML Programs

Are there any tools to transform programs written in an algebraic modeling language like GAMS,AMPL,... into a different formulation. E.g. there is a quadratic constraint $\sum_j b_i b_j = N, b \in \...
Lars Hadidi's user avatar
2 votes
1 answer
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Loglog transformation of optimization problem, how can the solution be equal to the nontransformed counterpart?

Consider the following two functions: $$y_t = e^{lt} \cdot e^{st} \cdot \prod_{p=0}^{n} x_{tp}^{b_{tp}}\tag1$$ Where $e^{lt}$ captures the trend, $e^{st}$ captures the seasonality and $x_{tp}$ is our ...
richardhansson's user avatar
5 votes
1 answer
432 views

Can we linearize the division of a binary variable by a continuous variable?

I'm trying to solve an MINLP problem where the following division term is appearing in the objective: $$z_r = \frac{x_{ry}}{\sum_r d_r x_{ry}}, \forall r, y,$$ where $x_{ry}$ is a 2D binary variable ...
Sourav Mondal's user avatar
3 votes
1 answer
187 views

If-then condition formulation to avoid variable multiplication

I'm trying to formulate the following logic: If $y_i =1$, then $c_i \leq x_i$ If $y_i =0$, then $c_i \leq 0$ Where $y_i$, $c_i$, and $x_i$ are decision variables. The easy way would be to write: $$c_i ...
Daniel Baquero's user avatar
2 votes
2 answers
374 views

Change the objective function formula change the complexity of a linear program?

I have a linear program, where I can use it with the same constraint to minimize objective 1 or minimize objective 2. I noted that when I use the formula of objective 2 the problem can be solved with ...
MAJID majid's user avatar
4 votes
1 answer
162 views

DCP formulation of sum of nonconvex and convex functions

I am trying to find a DCP formulation for the following convex objective function (using CVXPY): Let $x$ be the $N$-dimensional vector variable on which we optimize on, $c$ be a known scalar value ...
LowOdds's user avatar
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2 votes
2 answers
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Linearize a product of binary variables

I have a function to minimize which has the following term $$\sum_{i\in I}\sum_{j\in J}\sum_{k\in K}x_{ijk}N_{ij}a_{ijk},$$ where the variables are $x_{ijk}\in\{0,1\}$, $a_{ijk}$ are given as input ...
zdm's user avatar
  • 403
6 votes
1 answer
217 views

How to treat a system of bilinear constraints

A model contains constraints of the following form $R(k) \leq X(k) G(k)$ where $X(k)$ binary and $G(k)$, $R(k)$ non-negative variables. The index $k$ runs from $1$ to $50$. I linearise the equations ...
Clement's user avatar
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5 votes
2 answers
335 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 ...
Marry's user avatar
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4 votes
2 answers
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How to deal with an optimization problem that have a sum of nonlinear functions of Z as a constraint when Z is the quantity to be minimized?

I have to minimize a quantity $Z$ subject to the following constraints: $$ w_1 + w_2 + w_3 = 1 \tag{1}$$ $$ \frac{f_1(w_1 Z) + f_2(w_2 Z) + f_3(w_3 Z)}{Z} \ge k \tag{2}$$ where $k$ is a known ...
Green Noob's user avatar
7 votes
1 answer
380 views

Maximizing a Ratio/Percent

I'm using cvxpy to model a problem. Inside a very large and complex LP, I create two continuous, affine (unconstrained) expressions: $x$ and $y$. Due to how they ...
Adi Shavit's user avatar
4 votes
1 answer
284 views

Switching of decision variables to be equal to a certain decision variable according to a binary (indicator) variable

I would like to seek some advice on modeling the following: I have two integer decisions variables, $x, x'$, that are either equal or greater than zero and either of them is to be equated to a third ...
Mike's user avatar
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2 votes
1 answer
109 views

Formulating indicator constraint set

I am having difficulty formulating the indicator constraints for the following: Consider a set of $A_{n}$ decision variables such that $A_{1},A_{2},⋯,A_{n}<A$. While all of them are integers that ...
Mike's user avatar
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2 votes
1 answer
68 views

Reformulating to locate the second largest decision variable of a set of decision variables

Consider a set of $A_{vn}$ decision variables such that $A_{v1},A_{v2},\cdots,A_{vn}<A$. While this is the standard formulation finding the maximum value of $A_{vn}$, I would also like to find the ...
Mike's user avatar
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2 votes
1 answer
255 views

Linearize sum of continuous and boolean variable

For maximizing the objective function $\sum_i{d_i y_i}+ A x - B \cdot \mathbb{I}_{x>0}$, how can I linearize $ A x - B \cdot \mathbb{I}_{x>0}$ term where $d_i, A$ and $B$ are positive constants ...
Al Guy's user avatar
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6 votes
0 answers
138 views

Cases where RLT/SDP relaxation does not work well with standard quadratic optimization

(For people who don't know what RLT is): I am maximizing an indefinite quadratic function over a standard simplex, i.e., the standard quadratic optimization problem. A well-known approach is to relax ...
independentvariable's user avatar