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The center of the research of our institute is optimization. But what is optimization? Basically, optimization is the art of making good decisions. It provides us with a set of tools, mostly from the areas of computer science and mathematics, which are applicable in virtually all fields ranging from business, industry, biology, physics, medicine, data mining, engineering, to even art.

Every question that asks for a thing with a superlative feature is an optimization problem. Constructing the fastest car, finding the most profitable investment plan, finding a way to discover diseases as early and reliable as possible, scheduling your work so that you have the most spare time left for gaming – all of these are optimization tasks.

In this article, we will explore what an optimization problem in more detail, we will distinguish hard from easy problems and discuss what implications come from "hardness". We will discuss how exact and approximate algorithms tackle hard problems, give a slightly more formal definition of optimization problem, and list some more examples for optimization tasks.

Both our professional and private lives are sequences of decisions and each decision involves choosing between at least two options (otherwise we can hardly call it a decision). Amongst the possible choices, we search for the best, the optimal one. Any such search can be formulated as optimization problem, which has at least two major components:

  1. A set X of possible choices x∈X. X is called the solution space, since each choice x would be a potential "solution" to the problem. The elements x are called candidate solutions.
  2. Some way to compare two solutions x1,x2X and to tell which one is better.

1. What is Problem Hardness?

Naturally, we would like to find the best solution xX. There are quite a few problems where we can do this directly. For instance, if we have 100 coins and want to purchase as much cake as possible in a situation where on piece of cake costs 10 coins, we can directly and easily derive the solution x=10.

1.1. Time Complexity

Then there may be optimization problems where we cannot get the solution as result of directly solving a single equation, but by using an algorithm. Finding the shortest paths from one city to all n-1 other cities on a map is clearly an optimization task – which can directly be solved using Dijkstra's algorithm. Such algorithms need a certain number of "steps" to finish, and in case of a basic implementation of Dijkstra's algorithm, that number is roughly proportional to n2 (*). In other words, the more cities we have, the longer it will take to find all the shortest path. The needed time will grow roughly quadratically with the number of cities.

The relationship between the "size and features" of a problem and the time needed to solve it is called (time) complexity. For each problem, we want to use the best method (algorithm) to solve it. For different problems, there are different best algorithms and these have a different time complexity.

1.2. Simple Problems: Polynomial Time Complexity

Sorting a set of s numbers can be done in the scale of s*log(s) steps.

Assume we have map with e streets and v intersection. We know how many cars can go on each street per minute. The maximum amount of cars which can drive from a source intersection A to a target intersection B can be determined using the Edmonds-Karp algorithm within a number of steps roughly proportional to v*e2.

Clearly, the number of steps we need to get the solution of these problems depends on their size (called scale). The more streets we have on the map, the longer it takes to find how many cars we can send from A to B per time unit. The more elements we want to sort, the longer the sorting will take. As long as the number of required steps of an algorithm does not grow much faster than a reasonable polynomial, say n5, we can solve the problem by throwing hardware on it.

1.3. Hard Problems: Exponential Time Complexity

Unfortunately, there is a class of problems where the number of steps needed to (guarantee to get) get the exact globally best solution may rise exponentially with the problem scale (in the worst case)!

A log-log-scaled plot of different functions (including polynomially and exponentially growing ones as well as the factorial) to illustrate the speed of growth of these functions.
Figure 1: A log-log-scaled plot of different functions (including polynomially and exponentially growing ones as well as the factorial) to illustrate the speed of growth of these functions.

What this means is that if you have such a problem of a reasonable size and want to get the best-possible solution, you may need to wait far too long. And there are many such problems, including the following examples:

  • the Boolean Satisfiability Problem, asking whether a given Boolean formula can become true, which is quite important for circuit design, for instance,
  • the Bin Packing Problem, asking for the minimum number of containers (each of the same volume) needed to package a certain set of (differently-sized) items, which has quite obvious applications in logistics,
  • the Knapsack Problem, asking for the maximum sum value of items (selected from a specific set of items with different values and sizes) we can pack into a given volume, too, has quite a lot of obvious applications,
  • the Set Covering Problem, where a set of subsets is given and we want to find the minimum number of subsets whose union still covers all the elements in all subsets, which can be visualized as, e.g., the question where to put base stations in a mobile network or a situation where a company needs to purchase a certain set of supplies and different contractors offer different deals, each containing some of the supplies,
  • in Graph Coloring, we ask for the minimum number of colors to be assigned to the nodes of a given graph so that no two nodes connected by an edge have the same color, which has applications, for instance, in the register allocation during the code optimization of program compilation,
  • in the Quadratic Assignment Problem, we have n workshops that need to be assigned to n locations. We know the flows of goods between the workshops and the distances between the locations and want to find an assignment with minimal total sum product of flow*distance.
  • A huge variety of logistics problems, such as the Route Inspection Problem, the Vehicle Routing Problem, Arc Routing Problems as well as their versions with capacity limits, time limits, deadlines, penalties, peridocity, etc., are all hard.
  • Job Shop Scheduling, Flow Shop Scheduling, and their variants are hard tasks which ask for optimal assignments of tasks to machines in a factory.

In summary, many questions that we encounter in management, design, engineering, planning, and logistics are hard optimization problems. And we need to solve them.

2. How can we Solve Hard Problems?

2.1. Example: Traveling Salesman Problem

One of the most well-known hard combinatorial optimization problems is the Traveling Salesman Problem (TSP), which can be formulated as follows: Given is a set of n cities and the distances between each pair of two cities. What is the shortest possible tour that visits all cities and returns back to its starting point? A candidate solution is a permutation of the n cities, i.e., a sequence which contains each city exactly once. It describes the order in which the cities are to be visited and the way back is implied.

There are |X|=n! many such permutations, which turns out to be quite a large number even for moderate n as shown already in Figure 1, with 10! being 3'628'800 and 20!≈2.4×1018. It turns out that we do not yet have an exact algorithm which can solve any possible TSP in 1.9999n steps or less. And chances are that we will never have one.

2.2. Solving the Traveling Salesman Problem

But what does it mean that we cannot guarantee to find the solution of a TSP in reasonable time? First of all, the word "solution" is used here in a very strict and tight sense, in the sense of "exact" and "globally optimal": A solution to the TSP in this classical sense is only the shortest possible round-trip tour through the n cities.

But what if we widen this definition a little bit: Actually, every possible visiting sequence of the n cities which contains each city exactly once (and returns back to the first city) could be considered a solution. We then simply have solutions of different quality, which we thus call candidate solution.

Even if we have a TSP with 10000 cities, we actually can obtain one such solution very quickly, within n steps, simply by writing down all n cities directly in the same order as in the problem specification. Although there is a tiny chance that this might already be the exact best-possible solution to the TSP, it very, very likely is will be a very very bad (candidate) solution. Anyway, we thus have established we may not be able to guarantee to find the best possible solution for hard problems quickly, but we definitely can find some candidate solution.

2.3. The Two Poles of Problem Solving

We now know the two polar opposites of problem solving: On one hand – for most problems – we can find some random solution of poor quality quickly. On the other hand, finding the best-possible solution may take far too long to be feasible.

Such a situation is not satisfying and if we face it, what we would want is something in between.

A sketch of the time-quality trade-off we make with optimization on the example of random and optimal solutions for the Traveling Salesman Problem.
Figure 2: A sketch of the time-quality trade-off we make with optimization on the example of random and optimal solutions for the Traveling Salesman Problem.

2.4. Optimization: Getting Good Approximate Solutions in Reasonable Time

This "in between" is one of the things that optimization does. On the left end of the scale in Figure 2, we find the afore-mentioned random solutions which do not make use of any information from the problem. But we usually have a lot of information: In the TSP, for instance, we know the distances between the cities.

2.4.1. Constructive Heuristics

Constructive heuristics are methods which usually create a single candidate solution in a meaningful, one-shot fashion by utilizing information from the problem. Heuristics for the TSP, we could apply a nearest neighbor heuristic which creates a tour step-by-step by first picking a (random) city and then iteratively adding the city nearest to the previously added one to the tour. This means we need something in the scale of 0.5n2 steps, as every time we want to add a city, we have to check all not-yet added cites. It will normally produce much better solutions than creating random solutions, but usually also not find the best possible tour.

2.4.2. Local Search

If we move up the time scale, we arrive at the family of local search algorithms, which start with a single solution and try to iterative improve it. Assume we have created a solution for the TSP using a heuristic. We could now check whether we can swap two cities in the tour. If this results in a better tour, we take it. If not, we try again. Usually, there exist many different such refinement operators.

In the TSP, besides swapping cities, we could reverse a leg of the tour or rotate the cities a leg to the left or right.

Also, there exist several ways for applying the moves in each step of the local search: We could first try all n(n-1)/2 city pairs that can be swapped and actually swap the one which improves the tour the most, if any. Or we could try the swaps and pick the first improving swap.

A local search cannot guarantee that we find the optimal solution either. We may arrive in a dead end, a local optimum. If we use the city-swap operator, this could be a tour which cannot be improved by swapping two cities but needs a more complex change.

Thus, local search methods can be enhanced with a variety of additional measures, such as restarting if no improvement can be found (but, of course, remembering the best solution so far).

2.4.3. Population-Based Metaheuristics and their Hybrids

In order to prevent running into local optima to quickly, the concept of population-based metaheuristics exists, mainly known under the labels Evolutionary Computation, Evolutionary Algorithm, and Genetic Algorithm (which all have slightly different meanings).

Here, a set multiple of solutions in maintained and iteratively refined. However, these algorithms are usually way to slow to be of practical use. They are thus hybridized with local search: The resulting Memetic Algorithms can guard against getting stuck too early while having the speed of local search.

2.4.4. Exact Methods vs. Approximate Methods

Exact optimization algorithms naturally solve a problem to optimality, incurring the potential large, exponential runtime. It should be noted, however, that the runtime is only exponential in the worst case, e.g., in TSPs with particularly nasty distributions of cities. In many real problems, exact algorithms might be able to run in a reasonable time. The Concorde TSP Solver has solved a TSP with 85900 cities to optimality.

On the other hand, although metaheuristics and local search usually do not guarantee to find the optimal solution, they might actually do so, and they might do quite quickly. The LKH algorithm is a local search which has found the solution for the same 85900-city problem – but probably much faster than Concorde: To solve a 13'509 city-problem, Concorde needed three months on a cluster while LKH did the job in half a day on a desktop computer. Of course, not guarantee that the solution it found was optimal. We only know this because of Concorde.

Thus, both exact and approximate methods have drawbacks and advantages.

2.4.5. Everything is an Anytime Algorithm

Finally, it should be noted that most optimization methods are anytime algorithms. These are algorithms that can be stopped and provide an approximate solution at any time. This is easy to see for local search, for example: Once it has an initial solution, we can always stop it and just take the best solution it found so far. This is also true for all metaheuristics and, interestingly, also many exact methods.

Branch and bound algorithms iteratively and recursively divide the search space into regions and only search those which may improve upon the best solution they have so far. They repeat this until there is no such region left, which then means that the best-so-far solution is the best-possible solution. But, of course, we can stop them earlier and just take the best-so-far solution (losing the guarantee to get the best-possible solution in the process.

From this perspective, most optimization algorithms become comparable, processes of the development of solution quality over time …but this is another post altogether.

3. Formal Definition of Optimization Problem

So far, we have discussed roughly what an optimization problem is, a set of possible solutions X and a way to compare them. Then we have explored different types of algorithms for solving such problems. Let us now give a formal definition of optimization problems and some more examples.

3.1. Single-Objective Optimization Problem

How can we compare solutions? The most common way to do so is to define an objective function f which assigns a real value to each candidate solution x∈X, i.e., f:XR. f represents an optimization criterion, a quality or utility measure that we can understand. We call this basic problem type a single-objective optimization problem.

3.1.1. Minimization

Often, f is something that we want to minimize, maybe a cost or the amount of other resources which would be required to realize the solution. Then, candidate solution x1X is better than solution x2X if f(x1)<f(x2), worse if f(x1)>f(x2), and neither is better if f(x1)=f(x2).

Our goal would be to find the one solution xX with the smallest corresponding objective value f(x). Of course, there may be more than one such optimal solution. Thus, we search for xX with f(x)≥f(x)∀x∈X. If there is no other element x∈X with a smaller objective value than the one of x, then x is optimal – we cannot make a better choice.

3.1.2. Maximization

f could also be something that we want to maximize, maybe a profit that we could obtain when choosing the given solution or an accuracy, for instance. In this case, we look for elements xX with f(x)≤f(x)∀x∈X instead. Minimization and maximization tasks are two sides of the same coin, as we can turn any maximization problem fmax into a minimization task fmin by setting fmin(x)=-fmax(x).

3.2. More Examples

3.2.1. Travel to Shanghai

Let's say I want to travel from my home in Hefei to Shanghai. Obviously, there are several options to do so. Candidate solutions are, for example, x1="hard seat ticket for slower train K462", x2="second class seat in faster train D3016, x3="airplane MU5468 of the China Eastern Airlines", and so on. The space of solutions is then X=x1,x2,x3,…. As objective function f1, I could choose the required travel time in hours. It turns out that f(x1)≈5.4, f(x2)≈3.4, and f(x3)≈1.2. Amongst the three listed example choices, the airplane would be the best one – under criterion f1.

Since are only so-and-so many possible means of transportation available that I can use from my home in Hefei to Shanghai, the solution space x1 of this problem is finite. Problems with finite solution space are called combinatorial optimization problems.

The solution space X of this example has no clear structure and there are no direct relationships between the candidate solutions. There may be classes of solutions (such as "slow trains") but that's about all what we can see at first glance. Due to the lack of structure in the solution space, I would probably have to enumerate all possible solutions to find the optimum.

3.2.2. Function Minimization

Another, this time purely artificial, optimization task is to find the minimum of a five-dimensional variant of Ackley's function a(x):R5R with

The formula for Ackley's function.

The space of possible solutions here is a subset of the five-dimensional Euclidean space R5, e.g., X2=[-10,10]5. A candidate solution would be a five-dimensional real vector. If we want to find the minimum of a(x), then we can simply use it as objective function directly, i.e., set f(x)=a(x).

This time, we have uncountable infinitely many potential candidate solutions x∈X. This task thus is a so-called continuous or numerical optimization problem. Interestingly, X has a much clearer structure than the previous solution space: I can take an existing solution x1X and modify it a little bit, maybe by adding a vector of very small numbers, and get new, similar point x2 as result. If x2≈x1, then probably also f(x2)≈f(x1). This feature is called causality. It is quite important for optimization and I will maybe talk about it at another time. Furthermore, f is steady and symmetric. If we have two reasonably similar solutions (vectors) x1 and x2, we can combine them in a meaningful way in order to get another solution x3 which similar to both (and hopefully unites the different good characteristics of both) by setting x3=0.5(x1+x2). All of these properties may be useful when trying to minimize f2 and may help us to find the solution (which happens to be (0,0,0,0,0)T) in a finite amount of steps.

4. Optimization: A Huge Area

I hope that these discussions and examples have shown that optimization tasks can come in many different forms. Following the Law of the Instrument, virtually every task in mathematics and engineering, every question or decision in our daily life or work life can be considered as an optimization task. The way in which we solve these tasks will largely depend on the structures of the solution spaces X and the objective functions f (as well as on the time and computing resources we have available).

4.1. Objective Function Types

Some optimization tasks may have objective functions in a closed form, like Ackley's function. The stability of the architecture for a new 40 floor office building during an earthquake, however, can hardly be expressed as formula. For this task, as well as for many tasks from the engineering domain, the objective function may involve calling an external program running a randomized simulation. We may even need to simulate several different scenarios for each candidate architecture. Testing how good a given solution is may then take several minutes. Obviously, an objective function of this type will hardly be steady or differentiable, but it may still exhibit causality. It may still be able to solve it with the very same methods that we can use to find the minimum of Ackley's function.

4.2. Multi-Objective Optimization

Although optimization tasks are most commonly defined with a single goal, this is not necessarily always the case. Multi-objective tasks have n>1, i.e., multiple, objective functions. We can write them as vector function f:XRn with f=(f1,…,fn). When traveling from Hefei to Shanghai, I may not only be concerned with the travel time, but also with the travel cost. Hence, there are two conflicting goals: Traveling by airplane is fastest, but will also be more expensive. Then, finding candidate solutions representing suitable trade-offs between the different optimization criteria becomes our goal.

5. Summary

As you can see, optimization can be everything and anything. It is a really wide field, ranging from trivial problems over problems which we can solve with the right tools to tasks where we can only find approximate solutions in reasonable time. How we solve these problems depends on the structure of the solution space X as well as the features of the objective function(s) f as well as the algorithms that we are able to apply.

But this will be the topic of another post…

* Several theoretical aspects are represented in a very simplified (and thus not 100% correct) way in this article in order to ease reading and for the sake of brevity.