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How can I linearize or convexify this binary quadratic optimization problem?

I have an optimization problem as below. I am having a hard time with the last constraint. $\max \eta$ subject to ${\bf U}(:,m)^T{\bf A}{\bf U}(:,m)=0,m=1,2,\cdots,M$ here $\bf{A}$ is a Binary Matrix of size $N\times N$ (given, known) $\bf{U}$ is an…
KGM
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Forecasting with Holt's linear trend method

In Holt's linear trend method, given a time series $y_1,\cdots,y_t$, the forecasting equation at time $t+h$ based on data up to time $t$ is given by $$ \hat{y}_{t+h|t}= \ell_t + h b_t \tag{1} $$ where $\ell_t$ is an estimate of the level of the…
Kuifje
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BIP for Sudoku naturally integral?

I was reading through the following notes regarding solving a 9x9 Sudoku via a binary integer program https://vanderbei.princeton.edu/tex/talks/INFORMS_19/Sudoku.pdf The formulation is straightforward but on slide 10 the author claims that if the…
Ram
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What distuingishes a model-and-run framework solver from a programmatic solver?

I am looking for examples to understand what kind of benefits a model-and-run framework solver offers that a more programmatic solver doesn't do or doesn't do as well. What distinguishes a model-and-run framework? Context: I am trying to understand…
Geoffrey De Smet
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1 answer

Can't solve a specific instance of Gilmore & Gomory Cutting Stock with Delayed Column Generation

I am now somewhat comfortable with delayed column generation, but one specific example has been bugging me, as it's quite simple (on the second iteration we stop) and I can't reach the optimal solution. The instance has the following…
J. Dionisio
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Set constraint mip

I am programming a MIP problem. There are two continuous variables but only one can have a value and the other must then be zero. How do I give this as a constraint?
kc27
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Non-Integral Optimal Solutions of Totally Unimodular Linear Programs

If a Linear Program (LP) has Totally Unimodular constraint matrix, integer RHS vector, and has an optimal solution, then it has an integer optimal solution. But what about additional optimal solutions which are not integer? Do any mainstream LP…
Mark L. Stone
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Is there a better way of defining a constraint on positive integer variables such that no two variables are the same and are uniquely assigned a value

So suppose I have integer variables $x_1,x_2,\dots,x_N$ and I enforce that the integer variables are bounded i.e $1 \leq x_i \leq N$ I was interested in posing a constraint so that in the collection $\{ x_i\}$ I would see all values $1,\dots,N$.…
Vogtster
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Nonlinear optimization with constraint involving long product of optimization variables

I solve a nonlinear optimization problem of the form \begin{align} &\max x_0 \text{ such that } \\ &\left[ \sum_{j=0}^N \left(\alpha_j x_0^{j} \prod_{k=3}^j x_k \right)\right]^2 + \left[ \sum_{j=1}^N \left(\beta_j x_0^{j} \prod_{k=3}^j x_k…
Dan Doe
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Python vs. compiled languages in OR research using metaheuristics

In many articles that use metaheuristics to solve optimization problems, the programming language of choice is C++. For example, the following two articles present state-of-the-art metaheuristics to solve the Capacitated Vehicle Routing Problem and…
Leon Lan
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Training ML models to be used as objectives in optimization problems

Suppose that we have data (in my case, from a chemical process) which includes input data $X$ (characteristic of the material to be processed) and decision data $Y$ (decisions taken by operators to process this material), which produce an output…
Borelian
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How to find extreme rays

I am applying Benders decomposition and the dual is unbounded. I need to find the extreme rays to proceed, but I am not sure how to do that. Following is an example problem, can someone explain how the extreme rays (3,2) (1,3) were found in this…
John Bolton
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Books for integer and mixed integer programming

I would like to know which is a good theoretical book to study integer programming and mixed integer programming. Searching I found a large number of books, however each with different approaches. The approach I want to take is theoretical, starting…
David Morante
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What insights/info can be gathered from visualizing the Branch & Bound tree

What insights/info can be gathered from visualizing the Branch & Bound tree. Some commercial solvers (FICO Xpress) don't seem to provide any API to readily visualize the tree. It has to be implemented using callback functions.
user3711946
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Are "polynomial-time" algorithms for convex minimization actually pseudopolynomial time and/or FPTASes?

Motivating example This question concerns continuous convex minimization. However, the motivating example is the classic binary knapsack problem $$\text{maximize}\quad v^T x \qquad \text{subject to}\quad w^T x \leq W, ~~x_i \in \{0, 1\}.$$ There is…
Max
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