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# Discrete Optimization - Science topic

Explore the latest questions and answers in Discrete Optimization, and find Discrete Optimization experts.

Questions related to Discrete Optimization

I have a new idea (by a combination of a well-known SDP formulation and a randomized procedure) to introduce an approximation algorithm for the vertex cover problem (VCP) with a performance ratio of $2 - \epsilon$.

You can see the abstract of the idea in attached file and the last version of the paper in https://vixra.org/abs/2107.0045

I am grateful if anyone can give me informative suggestions.

I have two type of resources A and B. The resources are to be distributed (discretely) over k nodes.

the number of resources A is a

the number of resources B is b

resources B should be completely distributed (sum of resources taken by nodes should be b)

resources A may not be completely distributed over the nodes. In fact, we want to reduce the usage of resources A.

Given resources (A or B) to a node enhance the quality of the nodes, where the relation is non-linear.

All nodes should achieve a minimum quality.

What is the type of the problem and how I can find the optimal value?

Dear all,

I want to start learning discrete choice-based optimization so that I can use it later for my research works. I want to know about free courses, books, study materials available on this topic. Any suggestions will be appreciated.

Thanks,

Soumen Atta

What is stochastic and combinatorial optimization problem.

Also, How I can identify the problem i am working is

**Continuous or Discrete Optimization.**Hi,

I'm interested in solving a nonconvex optimization problem that contains continuous variables and categorical variables (e.g. materials) available from a catalog.

What are the classical approaches? I've read about:

- metaheuristics: random trial and error ;

- dimensionality reduction: https://www.researchgate.net/publication/322292981 ;

- branch and bound: https://www.researchgate.net/publication/321589074.

Are you aware of other systematic approaches?

Thank you,

Charlie

It seems that the quadprog function of MATLAB, the (conventional) interior-point algorithm, is not fully exploiting the sparsity and structure of the sparse QP formulation based on my results.

In Model Predictive Control, the computational complexity should scale linearly with the prediction horizon N. However, results show that the complexity scales quadratically with the prediction horizon N.

What can be possible explanations?

I would like to test the performance of a modified algorithm developed to solve a real-world problem that has these characteristics : (1)Discrete (2)Multi-Objective (3) Black-box (4)Large-scale.

How we can do this? and if there are no such test problems, is it sufficient to show its performance on the real-world problem only? (where the true Pareto Front is unknown)

Best regards,

In the

**mixed-variable heuristic optimization**domain, what is done when a**categorical**variable determines the existence of**continuous**or ordered**discrete**variables in each possible solution?To

**illustrate**, imagine an optimization problem to determine the**best tool to cut paper**.In this problem, a variable

**tool**can have the values "**knife**" or "**scissors**".- If its value is "
**scissors**", there's the continuous-valued**blade_size**variable. - If it's "
**knife**", there is the same**blade_size**continuous variable and also a**num_of_teeth**discrete variable

How can I deal with this problems using some metaheuristic designed to hadle categorical, continuous and discrete ordered variables?

My

**first tought**was to set the problem to the max possible dimensionality and, after choosing the value of the categorical variable, select (*if*commands) which other variables are going to be optimized and used to evaluate the solution.This probably will work, but it seems

**naive**to me. Do other more**sophisticated**methods to deal with this kind of problem exists? If yes, what are these methods?Dear scientists,

Hi. I am working on some

**dynamic network flow**problems with**flow-dependent transit times**in**system-optimal**flow patterns (such as the maximum flow problem and the quickest flow problem). The aim is to know how well existing algorithms handle actual network flow problems. To this end, I am in search of**realistic benchmark problems**. Could you please guide me to access such benchmark problems?Thank you very much in advance.

A trajectory is obtained for discrete points, what is the procedure for measuring the smoothness of this trajectory. The answer to this question will help me get a clear picture about the convergence rate of Legendre Pseudospectral method, where the rate of convergence is defined as 1/( N^2m/3−1 ). Here m is defined as the smoothness of the optimal trajectory and N is the number of nodes or points. This rate of convergence formula and further discussions can be found in the paper titled "

**Rate of convergence for the Legendre pseudospectral optimal control of feedback linearizable systems**" written by Wei Kang .Given a graph, I need to find a vertex (or set of vertices) that needs to be remove from this graph in order to reduce it's chromatic number.

I have started programming binary bat algorithm to solve knapsack problem. i have misunderstanding of position concept in binary space :

Vnew= Vold+(Current-Best) * f;

S= 1/ ( 1+Math.exp(-Vnew));

X(t+1) = { 1 S>Rnd , 0 Rnd>=S)

the velocity updating equation use both position from previous iteration (Current) and global best position (Best). In continuous version of BA, the position is real number but in the binary version, position of bat represented by binary number. In Knapsack Problem it means whether the item is selected or not. In the binary version, transfer function is used to transform velocity from real number to binary number. I'm confused whether the position in BBA is binary or real number ? if binary then the (Current-Best) can only be 1 - 0, 0 - 1, 1 - 1, etc. and if real number then how to get the continous representation if no continous equation to update the position (in original BA, the position updating equation is X(t+1) = X(t) + Vnew

Dear experts,

Hi. I appreciate any information (ideas, models, algorithms, references, etc.) you can provide to handle the following special problem or the more general problem mentioned in the title.

Consider a directed network G including a source s, a terminal t, and two paths (from s to t) with a common link e^c. Each Link has a capacity c_e and a transit time t_e. This transit time depends on the amount of flow f_e (inflow, load, or outflow) traversing e, that is, t_e = g_e(f_e), where the function g_e determines the relation between t_e and f_e. Moreover, g_e is a positive, non-decreasing function. Hence, how much we have a greater amount of flow in a link, the transit time for this flow will be longer (thus, the speed of the flow will be lower). Notice that, since links may have different capacities, thus, they could have dissimilar functions g_e.

The question is that:

How could we send D units of flow from s to t through these paths in the quickest time?

Notice: A few works have done [D. K. Merchant, et al.; M. Carey; J. E. Aronson; H. S. Mahmassani, et al.; W. B. Powell, et al.; B. Ran, et al.; E Köhler, et al.] relevant to dynamic networks with flow-dependent transit times. Among them, the works done by E Köhler, et al. are more appealing (at least for me) as they introduce models and algorithms based on the network flow theory. Although they have presented good models and algorithms ((2+epsilon)-approximate algorithms) for associated problems, I am looking for better results.

Almost all the optimization algorithms considers Function Evaluations to compare performance among various algorithms.

Do Function Evaluations number is the most important criteria? If yes/no why?

Hello,

Is it possible to mathematically model

**using***a binary variable***in the optimization problems?***continuous variables*For example, assume that

**'X'**is**{0,1}**. Can I define it as**0<=X<=1**in my problem and impose some additional constraints instead to force X to become only**'0'**or**'1'**?Regards,

Morteza

As there is no topology called best topology for all engineering Applications, I would like to study different network topologies applied in different engineering field. I would like to make worth discussion to analysis different topologies with advantages and disadvantages. Can one suggest good book to study different Network Topologies and their Applications in different Engineering?

Dear researchers,

I'm learning about data clustering, it presents a new area of research for me.

My questions are the following.

1. How can we formulate a data clustering problem as an optimization problem, in the other hand, how to construct good objective functions for data clustering ?

2 what is the best way to deal with data clustering problem ? (as a combinatorial problem or a continuous problem ).

Thank you for your consideration

Does any one know about the meaning of offset and the formula of this link: http://oeis.org/A002898/internal ?

After testing many instances I found out that when r = V / Vtotal <= ϕ (Golden Ratio) the algorithm takes a lot of time to printout the result.

When the ratio r is so close to ϕ , I noticed that : V / Vtotal = (V + Vtotal) / V (which represents the geometric relationship of the two quantities V and Vtotal in the Golden Ratio).

However, few of the instances having a ratio r > ϕ can take too long to print the results too.

So can this problem be related to ϕ or not?

PS: I got the idea of comparing it to ϕ after checking this answer Lower bound on running time for solving 3-SAT if P = NP

I have a all-node routing problem with non linear constraints

Hi, I am trying to implement Particle (or Genetical) Swarm Optimization. However, I am already stuck in the first step...

I am getting confused on how to initialise the particles, and what these particles (in terms of code) are.

Can anyone explain, please?

Thanks.

Andrea.

It's an efficient new hybrid meta-heuristic

– named in other context ANGEL – for solving discrete size optimization of truss structures. ANGEL combines ant colony optimization (ACO), genetic algorithm (GA) and local search (LS)

strategy. The procedures of ANGEL attempt to solve an optimization problem by repeating the following steps. First time, ACO searches the solution space and generates structure designs to provide the initial population for GA. After that, GA is executed and the pheromone set in ACO is updated when GA obtains a better solution. When GA terminates, ACO searches again by using the new pheromone set. ACO and GA search alternately and cooperatively in the solution space

I am trying to solve discrete and mixed variable optimization problems for the same I want to know the best constraint handling techniques. Which helps the problem to solve in minimum time.

ABSOLUTE VALUE OPERATOR LINEARIZATION

I have a nonlinear term in the objective function of my optimization problem as an Absolute Value function like |x-a|.

As far as I know, an Absolute Value operator makes the optimization problems nonlinear (i.e. NLP). How can I make it linear (LP or MILP)?

Max f(x)=g(x) + b*|x-a|

s.t. some linear constraints

Regards,

Morteza Shabanzadeh

Binary Variable * Real Variable = ?

1) lead to an equivalent 'Nonlinear' variable (and thus => MINLP),

2) lead to an equivalent 'Integer' variable, 'Discrete' I mean (and thus => MILP).

Which one is correct and why?

What is your idea to deal with this problem by adding a constraint and make the resultant problem MILP (if it is not MILP).

Regards,

Morteza Shabanzadeh

I will be grateful if anyone could suggest a reference where I can find a formal definition of “binary discrete optimization”

Hello,

I would like to know that how I can find the number of variables (especially the integer ones) in GAMS (General Algebraic Modeling System) codes.

Does GAMS platform have any options to show the number of variables?

Any help would be appreciated.

Regards,

Morteza

Hello,

As far as I know, the meta-heuristic algorithms such as GA, PSO, GSA, etc. generally find the optimal solution of 'unconstrained' optimization problems. If I have some constrains (equality and/or inequality equations), how will I be able to consider and model them in these kinds of algorithms?

I would greatly appreciate it if you kindly help me in this matter.

Regards,

Morteza Shabanzadeh

i.e in methods like window method using different windows, FFT, Fourier series method etc.

What are the total number of mathematical operations involved?

Some metaheuristics prove their superior performance in some kind of problems. Some of them are continuous optimization problems and others in discrete or binary optimization problems.

I am looking for new methods and new trends in optimization, which have absorbed many interests. I will be grateful to you if you can give me some information.

Suppose you are given a set of linear inequalities that define a polytope, and suppose for simplicity that each of the inequalities defines a facet. Now, suppose you wish to find a "small" collection of facets such that every vertex of the polytope lies on at least one of the facets in the collection. (As examples: the eight vertices of the cube can be covered by just two square facets, and the four vertices of the tetrahedron can be covered by two triangular facets.) This must be done without actually enumerating the vertices, which can be exponentially many. Are there any good exact or heuristic algorithms for this problem? (It can of course be viewed as a special kind of set covering problem, in which the elements to be covered are defined implicitly rather than given as an explicit list.)

The Convex Programing is a wide range field and uses many methods in order to solve every particular convex problem. After so many years of implementation, do you really believe that still exist any unsolved problem?

The discrete multi-criteria optimization problem. I'm looking for methods to determine weights in the weighted objective function. I would like to set the weights automatically taking into account the expert database of optimal cases.

It is the problem of optimizing the objective function. I think it's near topics: training with teacher and case based reasoning, but I found a few publications according to weights determination. Does anyone know previous studies on this topic?