A Generalized Quantum-Inspired Evolutionary Algorithm for Combinatorial Optimization Problems

Transcription

1 A Generalized Quantum-Inspired Evolutionary Algorithm for Combinatorial Optimization Problems Julio M. Alegría 1 julio.alegria@ucsp.edu.pe Yván J. Túpac 1 ytupac@ucsp.edu.pe 1 School of Computer Science Universidad Católica San Pablo (UCSP) Arequipa, Peru January 2014 ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 1/38

2 Table of contents 1 Introduction 2 QIEA 3 GQIEA 4 Case Study: Assignment Problem 5 Conclusions 6 References ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 2/38

3 Table of contents Introduction 1 Introduction 2 QIEA 3 GQIEA 4 Case Study: Assignment Problem 5 Conclusions 6 References ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 3/38

4 Introduction Introduction Optimize means finding the maximum or minimum of a certain function. When the domain of this function is finite we call it combinatorial optimization. Heuristics and metaheuristics are algorithms designed to provide sufficiently good solutions. Evolutionary Algorithms (EA) are metaheuristic optimization algorithms based on the principles of natural biological evolution. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 4/38

5 Introduction Introduction Advantages of EA: Robust and global. Highly Parallel. Easily adaptable. Disadvantages of EA: When validation is computationally expensive. Quantum-Inspired Evolutionary Algorithm (QIEA) [Han and Kim, 2002] addressed this problem by reducing the size of the population. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 5/38

6 Evolutionary Algorithms Introduction Evolutionary Algorithms Initial Population Create a random initial population X 0 Evaluation Compute the objective function f (x) of each candidate x i X t Fitness assignment From f (x) and X t, compute a( f (x i ),X t ) of each candidate x i Reproduction Create new individuals by genetic operators (crossover and mutation) Selection Select best individuals to reproduce. Figure 1: Basic cycle of an EA. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 6/38

7 Quantum Computing Introduction Quantum Computing Possibilities count, even if they never happen [Spector, 2004]. In Quantum Computing, the smallest unit of information is the Q-bit o qubit, which may be in state 0, 1 or in any superposition of the two [Hey, 1999]: Where: Ψ = α 0 + β 1 (1) α 2 is the probability of finding the Q-bit in state 0. β 2 is the probability of finding the Q-bit in state 1. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 7/38

8 Introduction Quantum-Inspired Algorithms Quantum-Inspired Algorithms Main problems in Quantum Computing: Difficulty of implementing a quantum computer. Difficulty of creating algorithms that take advantage of the processing power of these machines. Classical algorithms (capable of being run on classical computers) that use the principles of quantum computing, in order to improve their performance in problem solving [Moore and Narayanan, 1995]. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 8/38

9 Table of contents QIEA 1 Introduction 2 QIEA 3 GQIEA 4 Case Study: Assignment Problem 5 Conclusions 6 References ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 9/38

10 QIEA Quantum-Inspired Evolutionary Algorithm Proposed by Han, K.-H. and Kim, J.-H. on 2002 [Han and Kim, 2002]. QIEA, inspired on principles of quantum computing, proposes: A quantum-binary representation of individuals, using Q-bits. A variation operator called Q-gate. An observation process. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 10/38

11 Representation QIEA Representation A Q-bit is the smallest unit of information in QIEA, which is defined using a pair of numbers (α, β) as shown in equation 2. [ α β ] (2) A Q-bit individual is a string of Q-bits defined as shown in equation 3. [ α1 α 2... α m β 1 β 2... β m ] (3) ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 11/38

12 Representation QIEA Representation y [ α β ] β α x Figure 2: Graphical representation of a circle of unit radius with probability amplitudes to obtain the values 0 and 1 for any Q-bit. Source: [Abs da Cruz et al., 2007]. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 12/38

13 Operator QIEA Operator The update of the population is performed by a variation operator called Q-gate. This operator is defined as a rotation matrix as shown in equation 4. [ ] cos( θi ) sin( θ U( θ i ) = i ) (4) sin( θ i ) cos( θ i ) ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 13/38

14 Operator QIEA Operator y [ α k ] β k U( θi) [ ] αk β k θi x Figure 3: Graphic representation of the rotation of a Q-bit using the Q-gate operator. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 14/38

15 Algorithm QIEA Algorithm Algorithm 1 QIEA as proposed in [Han and Kim, 2002]. t 0 Initialize Q t while not stop-condition do t t + 1 Generate P t oberserving the states of Q t 1 Evaluate P t Store best solution in b t b t best between b t and b t 1 Q t Q-gate(Q t 1 ) end while ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 15/38

16 Algorithm QIEA Algorithm Initial Population Create an initial pop of individuals with equal probabilities. Sampling Observe the states of the population (classic individuals) Evaluation Compute the objective function of the classic individuals Variation Modify the population using Q-gate operator. Selection Pick and store the best classic individual of the. sampling Figure 4: Basic cycle of QIEA. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 16/38

17 Related work QIEA Related work [Chung et al., 2011] proposes a new initialization method for the QIEA, using priority lists to ensure greater diversity in the initial search area, applying the propose to unit commitment problems. In [Leon, 2012], a variation of the QIEA is presented which includes an elitism and mutation process of Q-bits. Also, a hill-climbing fase is performed at the end of each cicle. This variation is applied in multi-lateral negotiations on multi-agent systems. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 17/38

18 Related work QIEA Related work [Tsai et al., 2012] combines the principles of QIEA with the Latin squares to obtain a systematic reasoning mechanism to exploit the best solutions in a search space. [Ryu and Kim, 2012] presents a Multiobjective QIEA or MQIEA, which searches a pareto-optimal solution for multiobjective problems. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 18/38

19 Table of contents GQIEA 1 Introduction 2 QIEA 3 GQIEA 4 Case Study: Assignment Problem 5 Conclusions 6 References ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 19/38

20 GQIEA Generalized Quantum-Inspired Evolutionary Algorithm All these works maintain the quantum-binary representation of the QIEA. Disadvantages: Excessive increase of dimensionality. The need for a complex coding of solutions. GQIEA uses a, not quantum-binary, but quantum-combinatorial representation. This representation has the following advantages: Reduce the dimensionality of individuals. Simpler coding of individuals. Populations with less individuals, which increases efficiency and reduces the algorithm convergence time. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 20/38

21 Representation GQIEA Representation A generalized Q-bit or GQ-bit is the smallest unit of information in GQIEA, which is defined as a n size tuple, being n the total number of possible states S in which the GQ-bit can be found, as shown in equation 5. α 1 α 2. α n (5) ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 21/38

22 Representation GQIEA Representation A GQ-bit individual is a string of GQ-bits defined as shown in equation 6. α 1,1 α 1,2 α 1,m α 2,1 α 2,2 α 2,m (6) α n,1 α n,2 α n,m ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 22/38

23 Operators GQIEA Operators GQIEA also defines a variation operator called generalized Q-gate or GQ-gate. This operator is responsible of driving individuals toward optimal solutions. Two type of GQ-gate are proposed: Arithmetic GQ-gate. Geometric GQ-gate. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 23/38

24 Operators GQIEA Operators We have the j-th GQ-bit of a GQ-bit individual, that represents the superposition of n states S = {S 1, S 2,..., S n }. If the state S φ is the state present in the best individual found in the observation, the new probabilities of the GQ-bit will be determined as follows: Arithmetic GQ-gate { αi + α i = t si i = φ si i φ α i, i = 1, 2,..., n (7) Geometric GQ-gate { αi α i = t si i = φ si i φ α i, i = 1, 2,..., n (8) ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 24/38

25 Algorithm GQIEA Algorithm Algorithm 2 GQIEA proposed in this work. t 0 Initialize Q t while not stop-condition do t t + 1 Generate P t observing the states of Q t 1 Evaluate P t Store best solution in b t b t best between b t and b t 1 Q t GQ-gate(Q t 1, b t, t ) end while ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 25/38

26 Case Study: Assignment Problem Table of contents 1 Introduction 2 QIEA 3 GQIEA 4 Case Study: Assignment Problem 5 Conclusions 6 References ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 26/38

27 Case Study: Assignment Problem Case Study: Assignment Problem The Assignment Problem (AP) seeks the optimal match between two or more sets. Usually there are two sets: the tasks (actions to be assigned) and the agents (people or machines) [Pentico, 2007]. Task BGAP: Variation of the AP where multiple tasks can be assigned per agent, and where the goal is to minimize the maximum cost of the assignments made [Mazzola and Neebe, 1988]. Minimize max a A c a,t x a,t (9) t T ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 27/38

28 Case Study: Assignment Problem Case Study: Assignment Problem Using arithmetic GQ-gate Using arithmetic GQ-gate Figure 5: Comparison between the EA for Combinatorial Optimization and two instances of the GQIEA using arithmetic GQ-gate applied to AP (Task BGAP). ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 28/38

29 Case Study: Assignment Problem Case Study: Assignment Problem Using arithmetic GQ-gate Using arithmetic GQ-gate Algorithm(s) Parameter Value EA for Combinatorial Optimization Population size 100 EA for Combinatorial Optimization Selection method Roulette wheel EA for Combinatorial Optimization Crossover Uniforme EA for Combinatorial Optimization Probability of crossover 0.65 EA for Combinatorial Optimization Probability of mutation 0.08 GQIEA-5 Populaton size 5 GQIEA-10 Populaton size 10 GQIEA-5, GQIEA-10 GQ-Gate Arithmetic GQIEA-5, GQIEA Table 1: Parameters used in the algorithms in Figure 5. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 29/38

30 Case Study: Assignment Problem Case Study: Assignment Problem Using geometric GQ-gate Using arithmetic GQ-gate Figure 6: Comparison between the EA for Combinatorial Optimization and two instances of the GQIEA using geometric GQ-gate applied to AP (Task BGAP). ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 30/38

31 Case Study: Assignment Problem Case Study: Assignment Problem Using geometric GQ-gate Using arithmetic GQ-gate Algorithm(s) Parameter Value EA for Combinatorial Optimization Population size 100 EA for Combinatorial Optimization Selection method Roulette wheel EA for Combinatorial Optimization Crossover Uniforme EA for Combinatorial Optimization Probability of crossover 0.65 EA for Combinatorial Optimization Probability of mutation 0.08 GQIEA-5 Populaton size 5 GQIEA-10 Populaton size 10 GQIEA-5, GQIEA-10 GQ-Gate Geometric GQIEA-5, GQIEA Table 2: Parameters used in the algorithms in Figure 6. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 31/38

32 Table of contents Conclusions 1 Introduction 2 QIEA 3 GQIEA 4 Case Study: Assignment Problem 5 Conclusions 6 References ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 32/38

33 Conclusions Conclusions Results show that the arithmetic GQ-gate operator has a faster convergence than the geometric GQ-gate. The GQIEA has better results than traditional Combinatorial Optimization EA, even with a much smaller population. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 33/38

34 Table of contents References 1 Introduction 2 QIEA 3 GQIEA 4 Case Study: Assignment Problem 5 Conclusions 6 References ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 34/38

35 Referencias I References [Abs da Cruz et al., 2007] Abs da Cruz, A. V., Pacheco, M. A. C., and Vellasco, M. M. B. R. (2007). Quantum-Inspired Evolutionary Algorithms for Problems based on Numerical Representation. PhD thesis, Pontificia Universidade Catolica do Rio de Janeiro. [Chung et al., 2011] Chung, C., Yu, H., and Wong, K.-P. (2011). An advanced quantum-inspired evolutionary algorithm for unit commitment. Power Systems, IEEE Transactions on, 26(2): [Han and Kim, 2002] Han, K.-H. and Kim, J.-H. (2002). Quantum-inspired evolutionary algorithm for a class of combinatorial optimization. Evolutionary Computation, IEEE Transactions on, 6: [Hey, 1999] Hey, T. (1999). Quantum computing: an introduction. Computing & Control Engineering Journal, 10: ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 35/38

36 Referencias II References [Leon, 2012] Leon, F. (2012). Real-valued quantum-inspired evolutionary algorithm for multi-issue multi-lateral negotiation. In Intelligent Computer Communication and Processing (ICCP), 2012 IEEE International Conference on, pages [Mazzola and Neebe, 1988] Mazzola, J. and Neebe, A. (1988). Bottleneck generalized assignment problems. Engineering Costs and Production Economics, 14: [Moore and Narayanan, 1995] Moore, M. and Narayanan, A. (1995). Quantum-inspired computing. [Pentico, 2007] Pentico, D. W. (2007). Assignment problems: A golden anniversary survey. European Journal of Operational Research, 176: [Ryu and Kim, 2012] Ryu, S.-J. and Kim, J.-H. (2012). Distributed multiobjective quantum-inspired evolutionary algorithm (dmqea). In Robot Intelligence Technology and Applications 2012, pages ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 36/38

37 Referencias III References [Spector, 2004] Spector, L. (2004). Automatic Quantum Computer Programming, volume 7 of Genetic Programming. Springer US. [Tsai et al., 2012] Tsai, J.-T., Chou, J.-H., and Ho, W.-H. (2012). Improved quantum-inspired evolutionary algorithm for engineering design optimization. Mathematical Problems in Engineering, 1:27. ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 37/38

38 A Generalized Quantum-Inspired Evolutionary Algorithm for Combinatorial Optimization Problems Julio M. Alegría 1 julio.alegria@ucsp.edu.pe Yván J. Túpac 1 ytupac@ucsp.edu.pe 1 School of Computer Science Universidad Católica San Pablo (UCSP) Arequipa, Peru January 2014 ALEGRÍA, J.; TÚPAC, Y. (UCSP) January 2014, Pg 38/38