### mathematics and statistics online

Access provided by: anon Sign Out. A Review on Bilevel Optimization: From Classical to Evolutionary Approaches and Applications Abstract: Bilevel optimization is defined as a mathematical program, where an optimization problem contains another optimization problem as a constraint. These problems have received significant attention from the mathematical programming community.

Only limited work exists on bilevel problems using evolutionary computation techniques; however, recently there has been an increasing interest due to the proliferation of practical applications and the potential of evolutionary algorithms in tackling these problems. This paper provides a comprehensive review on bilevel optimization from the basic principles to solution strategies; both classical and evolutionary. A number of potential application problems are also discussed. As a realization of function optimization, we verify the correctness of the algorithm using numerical simulations of quantum circuits for the Knapsack problem.

Optimal flight gate assignment is a highly relevant optimization problem from airport management. Among others, an important goal is the minimization of the total transit time of the passengers. The corresponding objective function is quadratic in the binary decision variables encoding the flight-to-gate assignment. Hence, it is a quadratic assignment problem being hard to solve in general.

In this work we investigate the solvability of this problem with a D-Wave quantum annealer. These machines are optimizers for quadratic unconstrained optimization problems QUBO. Therefore the flight gate assignment problem seems to be well suited for these machines. We use real world data from a mid-sized German airport as well as simulation based data to extract typical instances small enough to be amenable to the D-Wave machine.

In order to mitigate precision problems, we employ bin packing on the passenger numbers to reduce the precision requirements of the extracted instances. We find that, for the instances we investigated, the bin packing has little effect on the solution quality. Hence, we were able to solve small problem instances extracted from real data with the D-Wave Q quantum annealer. Quantum annealing devices have been subject to various analyses in order to classify their usefulness for practical applications.

While it has been successfully proven that such systems can in general be used for solving combinatorial optimization problems, they have not been used to solve chemistry applications. In this paper we apply a mapping, put forward by Xia et al. Additionally we investigate the scaling in terms of needed physical qubits on a quantum annealer with limited connectivity. To the best of our knowledge, this is the first experimental study of quantum chemistry problems on quantum annealing devices. We find that current quantum annealing technologies result in an exponential scaling for such inherently quantum problems and that new couplers are necessary to make quantum annealers attractive for quantum chemistry.

Commercial quantum annealers from D-Wave Systems can find high quality solutions of quadratic unconstrained binary optimization problems that can be embedded onto its hardware.

This limitation poses a problem for using D-Wave machines to solve application-relevant problems, which can have thousands of variables. For the important Maximum Clique problem, this article investigates methods for decomposing larger problem instances into smaller ones, which can subsequently be solved on D-Wave. The reduction methods presented in this article include upper and lower bound heuristics in conjunction with graph decomposition, vertex and edge extraction, and persistency analysis. Recently, considerable attention has been paid to planning and scheduling problems for multiple robot systems MRS.

Such attention has resulted in a wide range of techniques being developed in solving more complex tasks at ever increasing speeds. At the same time, however, the complexity of such tasks has increased as such systems have to cope with ever increasing business requirements, rendering the above mentioned techniques unreliable, if not obsolete. Quantum computing is an alternative form of computation that holds a lot of potential for providing some advantages over classical computing for solving certain kinds of difficult optimization problems in the coming years.

Motivated by this fact, in this paper we demonstrate the feasibility of running a particular type of optimization problem on existing quantum computing technology.

The optimization problem investigate arises when considering how to optimize a robotic assembly line, which is one of the keys to success in the manufacturing domain. A small improvement in the efficiency of such an MRS can lead to huge saving in terms of time of manufacturing, capacity, robot life, and material usage. The nature of the quantum processor used in this study imposes the constraint that the optimization problem be cast as a quadratic unconstrained binary optimization QUBO problem.

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For the specific problem we investigate, this allows situations with one robot to be modeled naturally, meanwhile modeling the multi-robot generalization is less obvious and left as a topic for future research. The results show that for simple 1-robot tasks, the optimization problem can be straightforwardly solved within a feasible time span on existing quantum computing hardware.

### Correspondence Principles for Concave Orthogonal Games

The formulated CVRP is equipped with time-table which describes time-evolution of each vehicle. Therefore, various constraints associated with time can be successfully realized. Similarly, constraints of capacities are also introduced, where capacitated quantities are allowed to increase and decrease according to the cities in which vehicles arrive.

As a bonus of capacity-qubits, one can also obtain a description of state , which allows us to set various traveling rules, depending on the state of each vehicle. Rip-up and replace heuristic for finding a placement of constraints and embedding of variables in a hardware graph. As an alternative to updating constraint locations based on variable routing, simulated annealing or genetic algorithms can be used to modify placements.

## Duality (optimization) - Wikipedia

For example, define a gene to consist of a preferred location for each constraint, and a priority order for constraints. Given a gene, constraints are placed in order of priority, in their preferred location if it is available or the nearest available location otherwise. During simulated annealing, genes are mutated by perturbing the preferred location for a constraint or transposing two elements in the priority order. These algorithms tend to take much longer than rip-up-and-replace, but eventually produce very good embeddings. Owing to a limited number of qubits, it is often the case that a CSP or Ising model is too large to be mapped directly onto the hardware.

In this section, we describe two additional algorithms: divide-and-concur Gravel and Elser, ; Yedidia, , specialized to our case of Ising model energy minimization, and a new algorithm inspired by regional generalized belief propagation Yedidia et al. Here, z R is the subset of variables involved in the constraints of region R. Since embedding is slow in general, regions are fixed and embedded in hardware as a preprocessing step. At a high level, messages passed between regions indicate beliefs about the best assignments for variables, and these are used to iteratively update the biases on h R in hopes of converging upon consistent variable assignments across regions.

The two algorithms presented here implement this strategy in very different ways.

## Quantum Technology and Optimization Problems

Divide-and-concur DC Gravel and Elser, ; Yedidia, is a simple message passing algorithm that attempts to resolve discrepancies between instances of variables in different regions via averaging. In each region R , in addition to having an Ising model energy function E R z R representing its constraints, one introduces linear biases L R z R on its variables, initially set to 0. Let z i R denote the instance of variable z i in region R.

The two phases of each DC iteration are:. This basic algorithm tends to get stuck cycling between the same states; one mechanism to prevent this problem is to extend DC with difference map dynamics Yedidia, DC has been shown to perform well on constraint satisfaction problems and constrained optimization problems, and compared with other decomposition algorithms, has relatively low precision requirements for quantum annealing hardware.

On the other hand, like most decomposition algorithms, DC is not guaranteed to find a correct answer or even converge. The Boltzmann distribution is the unique minimum of the Helmholtz free energy. Our algorithm decomposes A into regional free energies. The resultant algorithm is similar in spirit to the generalized belief propagation algorithm of Yedidia et al. Sum—product belief propagation is related to critical points of the non-convex Bethe approximation, which for Ising energies reads.

The distribution p in the free energy is approximated by local beliefs marginals b i , b ij at each vertex and edge. In particular, if belief propagation converges then we have produced an interior stationary point of the constrained Bethe approximate free energy Yedidia et al. In exactly the same way as above, requiring consistent marginals induces a constrained minimization problem for this regional approximation.

The critical points of this problem are fixed points for a form of belief propagation. Specifically, for each variable z i in a constraint of R , the messages passed between variable and region are. For large regions, which involve a large number of variables, the first of these messages is intractable to compute. As in previous work Bian et al. In that work, the algorithm relied on minimizing the energy of the penalty model; here, we harness the ability of the hardware to sample from the low-energy configurations of the Ising model without relying on finding a ground state.

Unfortunately, it is not as simple as sampling from the Ising model formed from the constraints in a given region. Even if the hardware were sampling from its Boltzmann distribution, this would minimize the free energy of just that region. Only these variables gain corrective biases. Algorithm 2. Generalized belief propagation GBP based on regional decomposition. Beyond a proof of correctness, GBP offers a distinct computational advantage over our previous belief propagation algorithm from Bian et al. For ease of reference, we include the relevant message formulation from that work:.

This can be performed with a single programing call per region. Algorithm 2 is motivated by minimizing regional free energies, which is achieved at a Boltzmann distribution, and this is needed to prove soundness.

However, in practice, the ideal Boltzmann distribution is unnecessary. The computation of the messages uses the bitwise marginals of the distribution, and these can be very well approximated empirically from a modest sized sample from the low-energy spectrum.

## Mathematical optimization

We do expect that QA sampling can be Boltzmann-like as evidenced in Amin Small distortions to the energy spectrum, as indicated in that paper, should be averaged out in the computation of the marginals. One weakness in this algorithm is the need to know the temperature T in order to produce the corrective biases V i R z i. Benedetti et al. It seems likely that these techniques can be applied to GBP and will be incorporated into future work.

We apply the methods of the previous sections to solve problems in fault diagnosis, a large research area supporting an annual workshop since Our goal is to use fault diagnosis as an example of how to use the methods of this report, and we use these competitions as inspiration rather than adhere to their rules directly. The typical problem scenario is to inject a small number of faults into the circuit, using the specified fault modes for the targeted gates, and produce a number of input—output pairs.

Now, given only these input—output pairs as data, one wishes to diagnose the faulty gates that lead to these observations. Both the strong and weak fault model diagnosis problems are NP-hard. State-of-the-art performance for deterministic diagnosis is achieved by translating the problem into a SAT instance and using a SAT solver Metodi et al.

Greedy stochastic search produces excellent results in the weak fault model, but is less successful in the strong fault model Feldman et al. We study the effectiveness of the D-Wave hardware in two experiments. First, we examine the ability of the hardware to sample diverse solutions to a problem.