Semidefinite programming has been used successfully to build hierarchies of convex relaxations to approximate polynomial programs. This approach rapidly becomes computationally expensive and is often tractable only fo...
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ISBN:
(纸本)9783642208072
Semidefinite programming has been used successfully to build hierarchies of convex relaxations to approximate polynomial programs. This approach rapidly becomes computationally expensive and is often tractable only for problems of small sizes. We propose an iterative scheme that improves the semidefinite relaxations without incurring exponential growth in their size. The key ingredient is a dynamic scheme for generating valid polynomial inequalities for general polynomial programs. These valid inequalities are then used to construct better approximations of the original problem. As a result, the proposed scheme is in principle scalable to large general combinatorial optimization problems. For binarypolynomial programs, we prove that the proposed scheme converges to the global optimal solution for interesting cases of the initial approximation of the problem. We also present examples illustrating the computational behaviour of the scheme and compare it to other methods in the literature.
Hierarchies of semidefinite programs have been used to approximate or even solve polynomial programs. This approach rapidly becomes computationally expensive and is often tractable only for problems of small size. In ...
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Hierarchies of semidefinite programs have been used to approximate or even solve polynomial programs. This approach rapidly becomes computationally expensive and is often tractable only for problems of small size. In this paper, we propose a dynamic inequality generation scheme to generate valid polynomial inequalities for general polynomial programs. When used iteratively, this scheme improves the bounds without incurring an exponential growth in the size of the relaxation. As a result, the proposed scheme is in principle scalable to large general polynomialprogramming problems. When all the variables of the problem are non-negative or when all the variables are binary, the general algorithm is specialized to a more efficient algorithm. In the case of binarypolynomial programs, we show special cases for which the proposed scheme converges to the global optimal solution. We also present several examples illustrating the computational behavior of the scheme and provide comparisons with Lasserre's approach and, for the binary linear case, with the lift-and-project method of Balas, Ceria, and Cornu,jols.
Clock mesh is popular in high performance VLSI design because it is more robust against variations than clock tree at a cost of higher power consumption. In this paper, we propose novel techniques based on binary line...
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ISBN:
(纸本)9781424481927
Clock mesh is popular in high performance VLSI design because it is more robust against variations than clock tree at a cost of higher power consumption. In this paper, we propose novel techniques based on binary linear programming for clock mesh synthesis for the first time in the literature. The proposed approach can explore both regular and irregular mesh configurations, adapting to non-uniform load capacitance distribution. Our synthesis consists of two steps: mesh construction to minimize total capacitance and skew, and balanced sink assignment to improve slew/skew characteristics. We first show that mesh construction can be analytically formulated as binary polynomial programming (a class of nonlinear discrete optimization), then apply a compact linearization technique to transform into binary linear programming, significantly reducing computational overhead. Second, our balanced sink assignment enables a sink to tap the least loaded mesh segment (not the nearest one) with another binary linear programming which reduces both slew and skew. Experiments show that our techniques improve the worst skew and total capacitance by 14% and 15% over the state-of-the-art clock mesh algorithm [19] on ISPD09 benchmarks.
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