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Proving Strong Duality for Geometric Optimization Using a Conic Formulation
, 1999
"... Geometric optimization is an important class of problems that has many applications, especially in engineering design. In this article, we provide new simplified proofs for the wellknown associated duality theory, using conic optimization. After introducing suitable convex cones and studying their ..."
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Cited by 5 (1 self)
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Geometric optimization is an important class of problems that has many applications, especially in engineering design. In this article, we provide new simplified proofs for the wellknown associated duality theory, using conic optimization. After introducing suitable convex cones and studying their properties, we model geometric optimization problems with a conic formulation, which allows us to apply the powerful duality theory of conic optimization and derive the duality results valid for geometric optimization.
Efficient implementation of linear programming decoding
 IEEE Trans. Inform. Theory
, 2010
"... Linear programming (LP) decoding, originally proposed by Feldman et al. [4] as an approximation to the maximumlikelihood (ML) decoding of binary linear codes, solves a linear optimization problem formed by relaxing each of the finitefield paritycheck constraints into a number of linear constraint ..."
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Cited by 3 (2 self)
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Linear programming (LP) decoding, originally proposed by Feldman et al. [4] as an approximation to the maximumlikelihood (ML) decoding of binary linear codes, solves a linear optimization problem formed by relaxing each of the finitefield paritycheck constraints into a number of linear constraints. While providing a number of advantages over iterative messagepassing (IMP) decoders, such as its amenability to finitelength performance analysis, LP decoding is computationally more complex to implement in its original form than IMP decoding, due to both the large size of the relaxed LP problem and the inefficiency of using generalpurpose LP solvers. This paper explores ideas for fast LP decoding of lowdensity paritycheck (LDPC) codes. We first show a number of properties of the LP decoder, and by modifying the previously reported Adaptive LP decoding scheme [9] to allow removal of unnecessary constraints, we prove that LP decoding can be performed by solving a number of LP problems that contain at most one linear constraint derived from each of the paritycheck constraints. Then, as a step toward designing an efficient LP solver that takes advantage of the particular structure of LDPC codes, we study a sparse interiorpoint method for solving this sequence of linear programs. Since the most complex part of each iteration of the interiorpoint
A Conic Formulation for L P Norm Optimization
, 2000
"... In this paper, we formulate the l p norm optimization problem as a conic optimization problem, derive its standard duality properties and show it can be solved in polynomial time. We first define an ad hoc closed convex cone L p , study its properties and derive its dual. This allows us to express ..."
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Cited by 1 (0 self)
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In this paper, we formulate the l p norm optimization problem as a conic optimization problem, derive its standard duality properties and show it can be solved in polynomial time. We first define an ad hoc closed convex cone L p , study its properties and derive its dual. This allows us to express the standard l p norm optimization primal problem as a conic problem involving L p . Using convex conic duality and our knowledge about L p , we proceed to derive the dual of this problem and prove the wellknown regularity properties of this primaldual pair, i.e. zero duality gap and primal attainment. Finally, we prove that the class of l p norm optimization problems can be solved up to a given accuracy in polynomial time, using the framework of interiorpoint algorithms and selfconcordant barriers.
Deriving Duality for l_pnorm Optimization Using Conic Optimization
, 1999
"... In this paper, we formulate the l p norm optimization problem as a conic optimization problem and derive its standard duality properties. We first define an ad hoc closed convex cone L and derive its dual. We express then the standard l p norm optimization primal problem as a conic problem involvi ..."
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In this paper, we formulate the l p norm optimization problem as a conic optimization problem and derive its standard duality properties. We first define an ad hoc closed convex cone L and derive its dual. We express then the standard l p norm optimization primal problem as a conic problem involving L. Using convex conic duality, we derive the dual of this problem and prove the wellknown regularity properties of this primaldual pair, i.e. zero duality gap and dual attainment.
Decoding
"... Abstract—This paper explores ideas for fast linear programming (LP) decoding of lowdensity paritycheck (LDPC) codes. We first propose a modification of Adaptive LP decoding, and prove that it performs LP decoding by solving a number of linear programs that contain at most one linear constraint der ..."
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Abstract—This paper explores ideas for fast linear programming (LP) decoding of lowdensity paritycheck (LDPC) codes. We first propose a modification of Adaptive LP decoding, and prove that it performs LP decoding by solving a number of linear programs that contain at most one linear constraint derived from each of the paritycheck constraints. Then, as a step toward designing an efficient LP solver that exploits the structure of LDPC codes, we study a sparse interiorpoint implementation for solving this sequence of linear programs. Since the most complex part of each iteration of the interiorpoint algorithms is to solve a (usually illconditioned) system of linear equations for finding the step direction, we propose a framework for designing preconditioners to be used with the iterative methods for solving these systems. The effectiveness of the proposed approach is demonstrated via both analytical and simulation results. I.