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The Traveling Salesman Problem is perhaps the most studied of all combinatorial optimization problems. There has been a lot work in solving TSP instances to provable optimality. Concorde solves very large instances to provable optimality, with the largest solved having 85,900 points. There is a 100,000 city problem ... | The Traveling Salesman Problem (TSP) is indeed known for its complexity, and while much progress has been made in solving instances of varying sizes to optimality, not all instances are created equal in terms of difficulty.
Regarding your question about the smallest "difficult instance" of TSP, the instance with 6,73... | [
{
"from": "human",
"value": "The Traveling Salesman Problem is perhaps the most studied of all combinatorial optimization problems. There has been a lot work in solving TSP instances to provable optimality. Concorde solves very large instances to provable optimality, with the largest solved having 85,900 ... |
For some insertion-type heuristics for the traveling salesman problem, we have a fixed worst-case error bound of the form:
$$\frac{z^H}{z^*} \le \eta,$$
where $z^H$ is the objective value of the solution returned by the heuristic for a given instance, $z^*$ is the optimal objective value for that instance, and $\eta$ i... | As of my last knowledge update in October 2023, it remains true that the farthest insertion heuristic for the Traveling Salesman Problem (TSP) does not have a known fixed worst-case error bound of the form \(\frac{z^H}{z^*} \le \eta\) for all symmetric instances satisfying the triangle inequality.
While some heuristi... | [
{
"from": "human",
"value": "For some insertion-type heuristics for the traveling salesman problem, we have a fixed worst-case error bound of the form:\n$$\\frac{z^H}{z^*} \\le \\eta,$$\nwhere $z^H$ is the objective value of the solution returned by the heuristic for a given instance, $z^*$ is the optimal o... |
In a continuous-review $(r,Q)$ inventory system under a type-1 service level constraint, if the demand per unit time is distributed as $N(\mu,\sigma^2)$ and the lead time, $L$, is a constant, then the lead-time demand also has a normal distribution. The optimal reorder point (under some simplifying assumptions) is give... | You are correct in your assessment regarding the distribution of lead-time demand (LTD) when both demand and lead time are stochastic. In your scenario, demand per unit time \(D\) follows a normal distribution \(N(\mu, \sigma^2)\), and lead time \(L\) also follows a normal distribution \(N(\mu_L, \sigma_L^2)\).
Howev... | [
{
"from": "human",
"value": "In a continuous-review $(r,Q)$ inventory system under a type-1 service level constraint, if the demand per unit time is distributed as $N(\\mu,\\sigma^2)$ and the lead time, $L$, is a constant, then the lead-time demand also has a normal distribution. The optimal reorder point (... |
I want to solve a problem with docplex.cp (module for Constraint Programming optimizer CPLEX for Python), and the problem is available in .lp or .mps formats. But I did not find the attribute for reading a file in it.
For the CPLEX itself, utilizing the cplex module it can be done as following:
import cplex
Model = cp... | You're correct that the `docplex.cp` module does not support importing models directly from `.lp` or `.mps` files. The `import_model` method is specifically designed for models in CPO or FZN format.
However, if you're working with a problem defined in an `.lp` or `.mps` file and want to solve it using the CPLEX engin... | [
{
"from": "human",
"value": "I want to solve a problem with docplex.cp (module for Constraint Programming optimizer CPLEX for Python), and the problem is available in .lp or .mps formats. But I did not find the attribute for reading a file in it.\nFor the CPLEX itself, utilizing the cplex module it can be d... |
In the Flow/Job Shop problems, and other related scheduling problems, a common assumption is that at any given time, a particular job will be being processed on at most one machine (usually... none).
Now imagine a scheduling problem in which a particular job requires multiple "machines" simultaneously, e.g. multiple "r... | The scheduling problem you've described is indeed a variant of "resource constrained scheduling." In this scenario, each job requires multiple resources to be available simultaneously for it to be processed, which introduces complexity in the scheduling process.
In resource constrained scheduling, the challenge is to... | [
{
"from": "human",
"value": "In the Flow/Job Shop problems, and other related scheduling problems, a common assumption is that at any given time, a particular job will be being processed on at most one machine (usually... none).\nNow imagine a scheduling problem in which a particular job requires multiple \... |
Are there any classes of NP-hard combinatorial optimization problems where Second order cone programs (SOCP) gives a better approximation than linear programs (LP)?
I am looking for results in the flavor of Goemans and Williamson's celebrated result of approximating max-cut using semidefinite programs. But I want to u... | Yes, there are certain classes of NP-hard combinatorial optimization problems where second-order cone programming (SOCP) provides better approximation guarantees than linear programming (LP). While many famous approximation algorithms leverage semidefinite programming (SDP), similar approaches with SOCP have been explo... | [
{
"from": "human",
"value": "Are there any classes of NP-hard combinatorial optimization problems where Second order cone programs (SOCP) gives a better approximation than linear programs (LP)? \nI am looking for results in the flavor of Goemans and Williamson's celebrated result of approximating max-cut us... |
Does anyone know of any real-world examples in which supply chain contracts, of the type introduced by Pasternack (1985) and reviewed by Cachon (2003), are actually used in practice? I'm talking about contracts formulated as Stackelberg games, with a transfer payment between the two players, and expected profit functio... | There are indeed real-world examples of supply chain contracts that resemble the contractual frameworks discussed by Pasternack (1985) and reviewed by Cachon (2003). While specific case studies might not always explicitly label their arrangements in terms of Stackelberg games or Nash equilibrium, the fundamental concep... | [
{
"from": "human",
"value": "Does anyone know of any real-world examples in which supply chain contracts, of the type introduced by Pasternack (1985) and reviewed by Cachon (2003), are actually used in practice? I'm talking about contracts formulated as Stackelberg games, with a transfer payment between the... |
In Python's PuLP package, how can I get the value of a decision variable?
For example, if I declare a variable—
x = LpVariable('x_var', cat=LpContinuous)
—then solve the problem, then use
print(x)
Python just outputs:
x_var
How can I get the value of the variable, not its name?
| In the PuLP package for Python, after you solve your optimization problem, you can get the value of a decision variable by using the `.varValue` attribute of the variable. Here's how you can do it:
1. Define your variable.
2. Set up and solve your optimization problem.
3. Access the value using `.varValue`.
Here's an... | [
{
"from": "human",
"value": "In Python's PuLP package, how can I get the value of a decision variable?\nFor example, if I declare a variable—\nx = LpVariable('x_var', cat=LpContinuous)\n\n—then solve the problem, then use \nprint(x)\n\nPython just outputs:\nx_var\n\nHow can I get the value of the variable, ... |
There are two common approximations for the $(r,Q)$ inventory optimization problem that use the EOQ model. It is well known that one of them has a fixed worst-case error bound, but there is confusion in the literature about which one it is.
Approximation 1 (EOQ+SS Approximation): In this approximation, we set $Q$ using... | In the context of the two approximations for the \((r,Q)\) inventory optimization problem, the approximation that has a fixed worst-case error bound is **Approximation 1 (EOQ+SS Approximation)**.
To understand why, we need to analyze the nature of both approximations:
1. **EOQ+SS Approximation**:
- The approach is... | [
{
"from": "human",
"value": "There are two common approximations for the $(r,Q)$ inventory optimization problem that use the EOQ model. It is well known that one of them has a fixed worst-case error bound, but there is confusion in the literature about which one it is.\nApproximation 1 (EOQ+SS Approximation... |
When can we approximate a finite horizon MDP with infinite horizon?
Can we use infinite horizon stochastic shortest path problem on a directed acyclic graph?
| Approximating a finite horizon Markov Decision Process (MDP) with an infinite horizon MDP can be justified under certain conditions. Here are key scenarios where this approximation is reasonable:
1. **Stationarity**: If the transition probabilities and rewards are stationary (do not change over time), the finite horiz... | [
{
"from": "human",
"value": "\nWhen can we approximate a finite horizon MDP with infinite horizon? \nCan we use infinite horizon stochastic shortest path problem on a directed acyclic graph?\n\n"
},
{
"from": "gpt",
"value": "Approximating a finite horizon Markov Decision Process (MDP) with an i... |
I am aware that Benders Decomposition is readily available in CPLEX and in SCIP; but are there any (free) solvers that provide off the shelf stochastic programming MIP algorithms or a nice to work with programming environment ?
| Yes, there are several free solvers and environments that support stochastic programming and MIP algorithms. Here are a few notable options:
1. **COIN-OR (Computational Infrastructure for Operations Research)**:
- **Cbc**: The COIN-OR branch-and-cut solver can be used for MIP problems.
- **Clp**: For linear prog... | [
{
"from": "human",
"value": "I am aware that Benders Decomposition is readily available in CPLEX and in SCIP; but are there any (free) solvers that provide off the shelf stochastic programming MIP algorithms or a nice to work with programming environment ? \n"
},
{
"from": "gpt",
"value": "Yes,... |
What types of organizations/sponsors in the US support applied OR research? I'm familiar with NSF and NIH.
| In addition to the National Science Foundation (NSF) and the National Institutes of Health (NIH), there are several other organizations and sponsors in the United States that support applied operations research (OR) research. Here are some notable ones:
1. **Department of Defense (DoD)**: Various branches of the DoD, ... | [
{
"from": "human",
"value": "What types of organizations/sponsors in the US support applied OR research? I'm familiar with NSF and NIH.\n"
},
{
"from": "gpt",
"value": "In addition to the National Science Foundation (NSF) and the National Institutes of Health (NIH), there are several other organ... |
QUBO (Quadratic Unconstrained Binary Optimization) is the minimization of a quadratic function of binary variables.
It has been used for computer vision, Ramsey numbers, factoring numbers, the integer partitioning problem, the MaxCut problem, and many other problems.
But in the real-world, one would not factor numbers... | QUBO (Quadratic Unconstrained Binary Optimization) is indeed a versatile mathematical framework, and while it might not be the most efficient method for every specific problem, it has found applications in various real-world scenarios where its formulation aligns well with the structure of the problem at hand. Here are... | [
{
"from": "human",
"value": "QUBO (Quadratic Unconstrained Binary Optimization) is the minimization of a quadratic function of binary variables.\nIt has been used for computer vision, Ramsey numbers, factoring numbers, the integer partitioning problem, the MaxCut problem, and many other problems.\nBut in t... |
When dealing with mixed-integer-programs with many symmetric solutions it can take very long until the branch-and-bound-tree search is finished because symmetric optimal solutions cannot be pruned.
What techniques can I use for a MIP with symmetries to overcome these difficulties?
| Dealing with mixed-integer programs (MIPs) that have symmetric solutions can indeed complicate the branch-and-bound process, as the presence of symmetries can lead to an exponential number of equivalent solutions. Here are some techniques that can help manage symmetries and improve the efficiency of solving such proble... | [
{
"from": "human",
"value": "When dealing with mixed-integer-programs with many symmetric solutions it can take very long until the branch-and-bound-tree search is finished because symmetric optimal solutions cannot be pruned. \nWhat techniques can I use for a MIP with symmetries to overcome these difficult... |
Can someone please make a list of optimization problem libraries so that the community can add to and refine it?
I know a few off the top of my head.
| Sure! Here’s a list of some popular optimization problem libraries. Feel free to add more or refine the descriptions as needed:
### General Optimization Libraries
1. **SciPy** (Python)
- A scientific computing library that includes modules for optimization, including linear programming and nonlinear optimization.
... | [
{
"from": "human",
"value": "Can someone please make a list of optimization problem libraries so that the community can add to and refine it?\nI know a few off the top of my head.\n"
},
{
"from": "gpt",
"value": "Sure! Here’s a list of some popular optimization problem libraries. Feel free to ad... |
Suppose we have three variables, $x, y, z \in \mathbb R$. How can we linearize constraints with the following structure?
$$z \geq \min(x, y)$$
$$z \leq \max(x, y)$$
| To linearize the constraints \( z \geq \min(x, y) \) and \( z \leq \max(x, y) \) in a mathematical optimization context, we can make use of auxiliary variables and additional linear inequalities.
1. **Linearizing the minimum constraint \( z \geq \min(x, y) \)**:
The expression \( \min(x, y) \) can be rewritten usin... | [
{
"from": "human",
"value": "Suppose we have three variables, $x, y, z \\in \\mathbb R$. How can we linearize constraints with the following structure?\n$$z \\geq \\min(x, y)$$\n$$z \\leq \\max(x, y)$$\n"
},
{
"from": "gpt",
"value": "To linearize the constraints \\( z \\geq \\min(x, y) \\) and ... |
I am looking for (sources of) convex quadratic programming instances with linear constraints. I am open for both continuous and mixed integer problems, but do not want randomly generated instances.
I am aware of QPLIB, a collection of various types of convex/noncovex, integer/continuous, linearly constrained/quadrati... | If you're seeking sources for convex quadratic programming instances with linear constraints beyond QPLIB and the Maros and Meszaros test set, here are a few additional resources and collections you might find useful:
1. **Netlib Repository**: The Netlib repository contains a variety of test problems for linear and no... | [
{
"from": "human",
"value": "I am looking for (sources of) convex quadratic programming instances with linear constraints. I am open for both continuous and mixed integer problems, but do not want randomly generated instances. \nI am aware of QPLIB, a collection of various types of convex/noncovex, integer... |
When working on an optimization project, what is the typical time division (in percentage) between the various tasks that you have to work on:
Problem understanding/definition (figuring out what is to be
optimised, what are the relevant constraints, etc...)
Interfacing of the data needed to solve the problem (getting ... | The time division for various tasks in an optimization project can vary significantly depending on the specific project, its complexity, and the team's expertise. However, a general guideline for the allocation of time across the different phases might look something like this:
1. **Problem understanding/definition (3... | [
{
"from": "human",
"value": "When working on an optimization project, what is the typical time division (in percentage) between the various tasks that you have to work on:\n\nProblem understanding/definition (figuring out what is to be\noptimised, what are the relevant constraints, etc...)\nInterfacing of t... |
A list of OR Software forums would be nice.
Can someone import the list of OR Software forums from the (now) read-only OR Exchange at https://www.or-exchange.org/questions/9068/or-software-forums ?
Can you please delete extinct software, add additional software, and update links to be current?
| While I can't directly access external websites, I can help you compile a list of popular operations research (OR) software forums based on common knowledge. Below is an updated list of OR software forums, including both established and newer options. I've included brief descriptions of each software where applicable.
... | [
{
"from": "human",
"value": "A list of OR Software forums would be nice. \nCan someone import the list of OR Software forums from the (now) read-only OR Exchange at https://www.or-exchange.org/questions/9068/or-software-forums ?\nCan you please delete extinct software, add additional software, and update li... |
A logical constraint combines linear constraints using
logical operators, such as logical-and, logical-or, negation (that is, not), conditional statements (that is, if ... then ...) to express complex relations between linear constraints.
About chance constraints, it is desired to specify that a certain constraint be s... | Yes, there is a fundamental difference between logical constraints and chance constraints.
### Logical Constraints:
Logical constraints combine linear constraints using logical operators (such as AND, OR, NOT, and conditionals). They are used to express relationships between different constraints in a logical framewor... | [
{
"from": "human",
"value": "A logical constraint combines linear constraints using\nlogical operators, such as logical-and, logical-or, negation (that is, not), conditional statements (that is, if ... then ...) to express complex relations between linear constraints.\nAbout chance constraints, it is desire... |
I have found a solved example of A Stochastic Two-Period Model with No Setup Cost in the book Operational Research by Hillier, 7th edition, that has a lot of complicated calculations to arrive to the solution.
In the following example, $c=$cost of ordering, $h=$cost of holding, $p=$shortage cost and $^0_$ is the optima... | It sounds like you're dealing with a stochastic inventory management problem, particularly focusing on the two-period model. The key challenge in such models is often evaluating the cost functions that arise from the inventory policy, especially when they involve integrals and expected values.
In your case, you are tr... | [
{
"from": "human",
"value": "I have found a solved example of A Stochastic Two-Period Model with No Setup Cost in the book Operational Research by Hillier, 7th edition, that has a lot of complicated calculations to arrive to the solution.\nIn the following example, $c=$cost of ordering, $h=$cost of holding,... |
It is quite frequent that optimization algorithms have quite some parameters controlling their behavior (cooling schedule of a Simulated Annealing, length of the tabu list for a Tabu Search, population size for a Genetic Algorithm, etc...).
One will typically spend quite some time fine tuning those parameters using a "... | The idea of using instance characteristics to guide the selection of optimization algorithm parameters is often referred to as "parameter tuning based on instance features" or "instance-specific parameter tuning." This area has received attention in the literature, particularly in the context of various optimization al... | [
{
"from": "human",
"value": "It is quite frequent that optimization algorithms have quite some parameters controlling their behavior (cooling schedule of a Simulated Annealing, length of the tabu list for a Tabu Search, population size for a Genetic Algorithm, etc...).\nOne will typically spend quite some t... |
Is there any compact resource that includes a list of all academic journals in the OR/MS space, ranked by journal importance? Although there are some helpful features offered by publisher websites such as Elsevier journal finder, I think it would be helpful if such a list were to be added here as well.
| While there isn't a single definitive list of all academic journals in the Operations Research/Management Science (OR/MS) field ranked by importance, there are several resources and rankings that can provide useful insights. Here are a few you might find helpful:
1. **Journal Citation Reports (JCR)**: This resource pr... | [
{
"from": "human",
"value": "Is there any compact resource that includes a list of all academic journals in the OR/MS space, ranked by journal importance? Although there are some helpful features offered by publisher websites such as Elsevier journal finder, I think it would be helpful if such a list were t... |
It is well known that expectation preserves convexity: If $f(x)$ is convex and $Y$ is a random variable, then $\mathbb E[f(x-Y)]$ is convex. This property arises in, for example, inventory theory.
I have not been able to find a good source to cite for this well-known fact. Can anyone suggest one?
(For what it's worth, ... | The property you mentioned—that the expectation of a convex function preserves convexity—can be found discussed in various texts related to convex analysis, probability theory, and optimization. While you may not find a single source that explicitly states the result as you have described it, the underlying principles ... | [
{
"from": "human",
"value": "It is well known that expectation preserves convexity: If $f(x)$ is convex and $Y$ is a random variable, then $\\mathbb E[f(x-Y)]$ is convex. This property arises in, for example, inventory theory.\nI have not been able to find a good source to cite for this well-known fact. Can... |
I have recently learned about the logarithmic mean
$$\frac{x-y}{\ln(x)-\ln(y)},\quad x,y > 0.$$
It is used a lot in chemical engineering optimization models e.g. see slide 15 of Developing spatial branch & bound solvers.
But I wonder if anyone else has come across the logarithmic mean elsewhere in optimization?
Rao and... | The logarithmic mean is indeed a fascinating quantity with applications extending beyond its initial appearances in heat transfer and chemical engineering. Its definition, given by
$$ L(x, y) = \frac{x - y}{\ln(x) - \ln(y)}, \quad x, y > 0, $$
provides a way to measure the mean of two positive numbers in a manner t... | [
{
"from": "human",
"value": "I have recently learned about the logarithmic mean\n$$\\frac{x-y}{\\ln(x)-\\ln(y)},\\quad x,y > 0.$$\nIt is used a lot in chemical engineering optimization models e.g. see slide 15 of Developing spatial branch & bound solvers.\nBut I wonder if anyone else has come across the log... |
CPLEX solves non-convex quadratic problems to global optimality with a global optimality option (in version 12). The relevant pages are
this and this. I benchmarked many solvers, and see that CPLEX is by far the best solver in this problem. I am curious about the algorithms they use. Are these generally hidden, or can... | IBM CPLEX is a proprietary optimization solver, and while the specific algorithms used in CPLEX are not fully disclosed, the company does provide some information about the methods employed for solving various types of optimization problems, including non-convex quadratic problems.
In general, CPLEX uses a combinatio... | [
{
"from": "human",
"value": "CPLEX solves non-convex quadratic problems to global optimality with a global optimality option (in version 12). The relevant pages are \nthis and this. I benchmarked many solvers, and see that CPLEX is by far the best solver in this problem. I am curious about the algorithms th... |
I'm trying to use lazy constraints to solve an optimization problem. In some software such as CPLEX or GUROBI, they have some tools to handle them directly (in the original model) or using callback functions.
I was wondering what is the difference between them?
| Lazy constraints are a powerful technique in integer programming to improve the performance of optimization solvers by allowing them to handle constraints that only need to be enforced when they are violated by the current solution. The main difference between handling lazy constraints directly in the original model ve... | [
{
"from": "human",
"value": "I'm trying to use lazy constraints to solve an optimization problem. In some software such as CPLEX or GUROBI, they have some tools to handle them directly (in the original model) or using callback functions. \nI was wondering what is the difference between them?\n"
},
{
... |
The newer versions of Gurobi include a couple of built-in functionalities for multi-objective optimization: blended objectives and hierarchical (lexicographically ordered) objectives.
Suppose we wanted to minimize $2x + 3.5y + 7z$. Using the "blended" functionality by specifying weights for each objective, the relevan... | The two approaches you mentioned for specifying objectives in Gurobi—using blended objectives versus a manually constructed weighted-sum objective—essentially result in the same optimization problem if you're only working with the same weights.
However, there are some distinctions that might affect algorithmic and com... | [
{
"from": "human",
"value": "The newer versions of Gurobi include a couple of built-in functionalities for multi-objective optimization: blended objectives and hierarchical (lexicographically ordered) objectives. \nSuppose we wanted to minimize $2x + 3.5y + 7z$. Using the \"blended\" functionality by specif... |
Let's say I have an objective function
$$f(x_1,x_2, \cdots, x_n)$$
and $N$ constraints
$$x_i \ge 0. $$
I am trying to solve it with KKT conditions. Now the objective function becomes
$$f(x_1,x_2, \cdots, x_n)+ \mu_i(g_i(x)).$$
I want to solve it using a C program so I solved the equation manually for both cases where... | When you are using the Karush-Kuhn-Tucker (KKT) conditions to solve an optimization problem with constraints, particularly inequality constraints, the approach you should take involves the complementary slackness condition. The complementary slackness states that:
$$ \mu_i g_i(x) = 0 \quad \text{for all } i $$
Where ... | [
{
"from": "human",
"value": "Let's say I have an objective function \n$$f(x_1,x_2, \\cdots, x_n)$$\nand $N$ constraints \n$$x_i \\ge 0. $$\nI am trying to solve it with KKT conditions. Now the objective function becomes\n$$f(x_1,x_2, \\cdots, x_n)+ \\mu_i(g_i(x)).$$\nI want to solve it using a C program so ... |
Qualitatively, in my experience in the solving of large scale MILPs, it is common that binary variables corresponding to "edge possibility" components are frequently chosen. Intuitively, these seem associated with various "critical values" within the problem domain.
This seems to resonate with the approach of Lagrange ... | Using Lagrange multipliers in the context of Mixed Integer Linear Programming (MILP) can be quite insightful, particularly when dealing with large-scale problems that contain a mix of continuous and binary variables. While traditional Lagrange multiplier techniques are primarily associated with continuous optimization ... | [
{
"from": "human",
"value": "Qualitatively, in my experience in the solving of large scale MILPs, it is common that binary variables corresponding to \"edge possibility\" components are frequently chosen. Intuitively, these seem associated with various \"critical values\" within the problem domain.\nThis se... |
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