1,241 research outputs found
Tight Bound for Sum of Heterogeneous Random Variables: Application to Chance Constrained Programming
We study a tight Bennett-type concentration inequality for sums of
heterogeneous and independent variables, defined as a one-dimensional
minimization. We show that this refinement, which outperforms the standard
known bounds, remains computationally tractable: we develop a polynomial-time
algorithm to compute confidence bounds, proved to terminate with an
epsilon-solution. From the proposed inequality, we deduce tight
distributionally robust bounds to Chance-Constrained Programming problems. To
illustrate the efficiency of our approach, we consider two use cases. First, we
study the chance-constrained binary knapsack problem and highlight the
efficiency of our cutting-plane approach by obtaining stronger solution than
classical inequalities (such as Chebyshev-Cantelli or Hoeffding). Second, we
deal with the Support Vector Machine problem, where the convex conservative
approximation we obtain improves the robustness of the separation hyperplane,
while staying computationally tractable
Stochastic graph partitioning: quadratic versus SOCP formulations
International audienceWe consider a variant of the graph partitioning problem involving knapsack constraints with Gaussian random coefficients. In this new variant, under this assumption of probability distribution, the problem can be traditionally formulated as a binary SOCP for which the continuous relaxation is convex. In this paper, we reformulate the problem as a binary quadratic constrained program for which the continuous relaxation is not necessarily convex. We propose several linearization techniques for latter: the classical linearization proposed by Fortet (Trabajos de Estadistica 11(2):111โ118, 1960) and the linearization proposed by Sherali and Smith (Optim Lett 1(1):33โ47, 2007). In addition to the basic implementation of the latter, we propose an improvement which includes, in the computation, constraints coming from the SOCP formulation. Numerical results show that an improvement of SheraliโSmithโs linearization outperforms largely the binary SOCP program and the classical linearization when investigated in a branch-and-bound approach
ํ๋ฅ ์ต๋ํ ์กฐํฉ์ต์ ํ ๋ฌธ์ ์ ๋ํ ๊ทผ์ฌํด๋ฒ
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ผ๋ฌธ(์์ฌ)--์์ธ๋ํ๊ต ๋ํ์ :๊ณต๊ณผ๋ํ ์ฐ์
๊ณตํ๊ณผ,2019. 8. ์ด๊ฒฝ์.In this thesis, we consider a variant of the deterministic combinatorial optimization problem (DCO) where there is uncertainty in the data, the probability maximizing combinatorial optimization problem (PCO). PCO is the problem of maximizing the probability of satisfying the capacity constraint, while guaranteeing the total profit of the selected subset is at least a given value. PCO is closely related to the chance-constrained combinatorial optimization problem (CCO), which is of the form that the objective function and the constraint function of PCO is switched. It search for a subset that maximizes the total profit while guaranteeing the probability of satisfying the capacity constraint is at least a given threshold. Thus, we discuss the relation between the two problems and analyse the complexities of the problems in special cases. In addition, we generate pseudo polynomial time exact algorithms of PCO and CCO that use an exact algorithm of a deterministic constrained combinatorial optimization problem. Further, we propose an approximation scheme of PCO that is fully polynomial time approximation scheme (FPTAS) in some special cases that are NP-hard. An approximation scheme of CCO is also presented which was derived in the process of generating the approximation scheme of PCO.๋ณธ ๋
ผ๋ฌธ์์๋ ์ผ๋ฐ์ ์ธ ์กฐํฉ ์ต์ ํ ๋ฌธ์ (deterministic combinatorial optimization problem : DCO)์์ ๋ฐ์ดํฐ์ ๋ถํ์ค์ฑ์ด ์กด์ฌํ ๋๋ฅผ ๋ค๋ฃจ๋ ๋ฌธ์ ๋ก, ์ด ์์ต์ ์ฃผ์ด์ง ์์ ์ด์์ผ๋ก ๋ณด์ฅํ๋ฉด์ ์ฉ๋ ์ ์ฝ์ ๋ง์กฑ์ํฌ ํ๋ฅ ์ ์ต๋ํํ๋ ํ๋ฅ ์ต๋ํ ์กฐํฉ ์ต์ ํ ๋ฌธ์ (probability maximizing combinatorial optimization problem : PCO)์ ๋ค๋ฃฌ๋ค. PCO์ ๋งค์ฐ ๋ฐ์ ํ ๊ด๊ณ๊ฐ ์๋ ๋ฌธ์ ๋ก, ์ด ์์ต์ ์ต๋ํํ๋ฉด์ ์ฉ๋ ์ ์ฝ์ ๋ง์กฑ์ํฌ ํ๋ฅ ์ด ์ผ์ ๊ฐ ์ด์์ด ๋๋๋ก ๋ณด์ฅํ๋ ํ๋ฅ ์ ์ฝ ์กฐํฉ ์ต์ ํ ๋ฌธ์ (chance-constrained combinatorial optimization problem : CCO)๊ฐ ์๋ค. ์ฐ๋ฆฌ๋ ๋ ๋ฌธ์ ์ ๊ด๊ณ์ ๋ํ์ฌ ๋
ผ์ํ๊ณ ํน์ ์กฐ๊ฑด ํ์์ ๋ ๋ฌธ์ ์ ๋ณต์ก๋๋ฅผ ๋ถ์ํ์๋ค. ๋ํ, ์ ์ฝ์์ด ํ๋ ์ถ๊ฐ๋ DCO๋ฅผ ๋ฐ๋ณต์ ์ผ๋ก ํ์ด PCO์ CCO์ ์ต์ ํด๋ฅผ ๊ตฌํ๋ ์ ์ฌ ๋คํญ์๊ฐ ์๊ณ ๋ฆฌ์ฆ์ ์ ์ํ์๋ค. ๋ ๋์๊ฐ, PCO๊ฐ NP-hard์ธ ํน๋ณํ ์ธ์คํด์ค๋ค์ ๋ํด์ ์์ ๋คํญ์๊ฐ ๊ทผ์ฌํด๋ฒ(FPTAS)๊ฐ ๋๋ ๊ทผ์ฌํด๋ฒ์ ์ ์ํ์๋ค. ์ด ๊ทผ์ฌํด๋ฒ์ ์ ๋ํ๋ ๊ณผ์ ์์ CCO์ ๊ทผ์ฌํด๋ฒ ๋ํ ๊ณ ์ํ์๋ค.Chapter 1 Introduction 1
1.1 Problem Description 1
1.2 Literature Review 7
1.3 Research Motivation and Contribution 12
1.4 Organization of the Thesis 13
Chapter 2 Computational Complexity of Probability Maximizing Combinatorial Optimization Problem 15
2.1 Complexity of General Case of PCO and CCO 18
2.2 Complexity of CCO in Special Cases 19
2.3 Complexity of PCO in Special Cases 27
Chapter 3 Exact Algorithms 33
3.1 Exact Algorithm of PCO 34
3.2 Exact Algorithm of CCO 38
Chapter 4 Approximation Scheme for Probability Maximizing Combinatorial Optimization Problem 43
4.1 Bisection Procedure of rho 46
4.2 Approximation Scheme of CCO 51
4.3 Variation of the Bisection Procedure of rho 64
4.4 Comparison to the Approximation Scheme of Nikolova 73
Chapter 5 Conclusion 77
5.1 Concluding Remarks 77
5.2 Future Works 79
Bibliography 81
๊ตญ๋ฌธ์ด๋ก 87Maste
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