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...twenty years. Semi-definite Programming can be used to model many practical problems in vary fields such as Convex constrained Optimization, Combinatorial Optimization, Control Theory,... We refer to =-=[32]-=- for a general survey and applications of SDP. Algorithms for solving SDP have been explosively studied since a major works are made by Nesterov and Nemirovski [18], [19], [20], [21], in which they sh...

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...ings are used in context of Linear Matrix Inequalities constraints (LMIs), see [28], [31]. LMIs can also be solved numerically by recent interior point methods for semi-definite programming, see [6], =-=[26]-=-. ∗Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, (email: glan@ise.ufl.edu). 1 Linear Programming is a special case of Semidefinite Programming, as wel...

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...d the growth condition of the objective function. We now ready to describe our non-smooth algorithm as follows. Each main Step (Step 1), to obtain the new iterate, we run the sub-gradient method (see =-=[23]-=-) for K = 4M2µ2, where µ is defined in (2.5), with the input is the current iterate. In other words, we restart the sub-gradient algorithm after a constant number K = 4M2µ2 of iterations. We denote {x...

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...onstraints increase because of computational cost per each iteration. Recently, first-order methods are focused because of the efficiency in solving large scale SDP such as Nesterov’s optimal methods =-=[24]-=-, [25], Nemirovski’s prox-method [17] and spectral bundle methods [5]. In system and control theory, system identification and signal processing, Semi-definite Programmings are used in context of Line...

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...n that in view of Assumption 1, the pair of primal and dual SDP problem (2.1) and (2.2) satisfy the Slater’s condition, hence they have optimal solutions and their associated gap duality is zero, see =-=[1]-=-. Moreover, a primal-dual optimal solution of (2.1) and (2.2) can be found by solving the complementarity problem as following Linear Matrix Inequalities constraints Ax ≤ B AT y = c y ≤ 0 〈x, ...

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...grammings are used in context of Linear Matrix Inequalities constraints (LMIs), see [28], [31]. LMIs can also be solved numerically by recent interior point methods for semi-definite programming, see =-=[6]-=-, [26]. ∗Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, (email: glan@ise.ufl.edu). 1 Linear Programming is a special case of Semidefinite Programming, ...

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...Recently, first-order methods are focused because of the efficiency in solving large scale SDP such as Nesterov’s optimal methods [24], [25], Nemirovski’s prox-method [17] and spectral bundle methods =-=[5]-=-. In system and control theory, system identification and signal processing, Semi-definite Programmings are used in context of Linear Matrix Inequalities constraints (LMIs), see [28], [31]. LMIs can a...

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...t role in algorithmic convergence proofs. In particular, Luo and Tseng showed the power of error bound idea in deriving the linear convergent rate in many algorithm for variety class of problems, see =-=[16]-=-, [15], [30]. However, it is not easy to obtain an error bound except in linear and quadratic cases, or when the Slater constraint qualification condition holds, see [3]. In [33], Zhang derived error ...

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... from a arbitrary point to the feasible solution set of (6.34). The minimum constant LH satisfies the growth condition (6.35) is called the Hoffman constant which is well studied in [33], [27], [13], =-=[4]-=- and [34]. That constant can be easily estimated in some cases, especially in linear system of equations. In that case, the Hoffman constant is the smallest non-zero singular value of the matrix A. Th...

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...ational cost per each iteration. Recently, first-order methods are focused because of the efficiency in solving large scale SDP such as Nesterov’s optimal methods [24], [25], Nemirovski’s prox-method =-=[17]-=- and spectral bundle methods [5]. In system and control theory, system identification and signal processing, Semi-definite Programmings are used in context of Linear Matrix Inequalities constraints (L...

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... 2: Go to Step 1. The convergence property is described in following Theorem. Theorem 15 For any k ≥ 1, f(xk) ≤ 1 2 f(xk−1). 16 Proof. By convergence properties of Nesterov accelerate gradient method =-=[22]-=-, we have f(xk)− f∗ ≤ 4‖A‖ 2 (K + 2)2 ‖xk−1 − x∗‖2. Using the Hoffman error bound and note that f∗ = 0, we obtain f(xk) ≤ 4‖A‖ 2L2H (K + 2)2 f(xk−1) ≤ 1 2 f(xk−1), where the second inequality is follo...

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...nds for general convex conic problem under some various conditions. The error bound for Semidefinite Programming was studied by Deng and Hu in[3], Jourani and Ye in[8]. Related topics can be found in =-=[27]-=-, [29], [14]. The paper is organized as follows. In Section 2, we introduce the problem of interest and the Slater constraint constraint qualification condition is made. Respectively, in Section 3 and...

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...en because of the convexity of the objective function, it is easy to see that x∗ is a feasible solution to (5.16) for every step, i.e. x∗ ∈ Lt, t = 1, 2, ..., T − 1. Furthermore, using the Lemma 1 in =-=[9]-=- and the subproblem (5.16), we have ‖xτ − x∗‖2 + ‖xτ−1 − xτ‖2 ≤ ‖xτ−1 − x∗‖2, τ = 1, 2, ..., T. 12 Summing up the above inequalities we obtain ‖xT − x∗‖2 + T∑ τ=1 ‖xτ−1 − xτ‖2 ≤ ‖x0 − x∗‖2,∀x∗ ∈ X∗. T...

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... made by Nesterov and Nemirovski [18], [19], [20], [21], in which they showed that Interior Point (IP) methods for Linear Programming (LP) can be extended to SDP. Related topics can be found in [29], =-=[14]-=-. Despite the fact that SDP can be solved in polynomial time by IP methods, they become impractical when the number of constraints increase because of computational cost per each iteration. Recently, ...

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...ints increase because of computational cost per each iteration. Recently, first-order methods are focused because of the efficiency in solving large scale SDP such as Nesterov’s optimal methods [24], =-=[25]-=-, Nemirovski’s prox-method [17] and spectral bundle methods [5]. In system and control theory, system identification and signal processing, Semi-definite Programmings are used in context of Linear Mat...

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...an be applied for solving LP. In this paper, we propose a linearly convergent algorithm for Linear system of inequalities, which require a weaker assumption than the one for LMIs problem. We refer to =-=[12]-=- for other linear convergent algorithms for Linear system of inequalities. Error bounds usually play an important role in algorithmic convergence proofs. In particular, Luo and Tseng showed the power ...

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...ontrol Theory,... We refer to [32] for a general survey and applications of SDP. Algorithms for solving SDP have been explosively studied since a major works are made by Nesterov and Nemirovski [18], =-=[19]-=-, [20], [21], in which they showed that Interior Point (IP) methods for Linear Programming (LP) can be extended to SDP. Related topics can be found in [29], [14]. Despite the fact that SDP can be solv...

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...ion, Control Theory,... We refer to [32] for a general survey and applications of SDP. Algorithms for solving SDP have been explosively studied since a major works are made by Nesterov and Nemirovski =-=[18]-=-, [19], [20], [21], in which they showed that Interior Point (IP) methods for Linear Programming (LP) can be extended to SDP. Related topics can be found in [29], [14]. Despite the fact that SDP can b...

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...gorithmic convergence proofs. In particular, Luo and Tseng showed the power of error bound idea in deriving the linear convergent rate in many algorithm for variety class of problems, see [16], [15], =-=[30]-=-. However, it is not easy to obtain an error bound except in linear and quadratic cases, or when the Slater constraint qualification condition holds, see [3]. In [33], Zhang derived error bounds for g...

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...y,... We refer to [32] for a general survey and applications of SDP. Algorithms for solving SDP have been explosively studied since a major works are made by Nesterov and Nemirovski [18], [19], [20], =-=[21]-=-, in which they showed that Interior Point (IP) methods for Linear Programming (LP) can be extended to SDP. Related topics can be found in [29], [14]. Despite the fact that SDP can be solved in polyno...

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...re ‖A‖ denotes the operator norm of A with respected to the pair of norm ‖.‖2 and ‖.‖F defined as follows ‖A‖ := max{‖Au‖∗F : ‖u‖ ≤ 1} (4.10) Proof. The proof immediately follows the Proposition 1 of =-=[11]-=-, in which U = U∗ = Rn, V = V ∗ = Sn, and ψ = (distSn − )2, where distSn − is the distance function to the cone Sn− measured in terms of the norm ‖.‖F . Note that distSn − is a convex function with 2-...

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...ks are made by Nesterov and Nemirovski [18], [19], [20], [21], in which they showed that Interior Point (IP) methods for Linear Programming (LP) can be extended to SDP. Related topics can be found in =-=[29]-=-, [14]. Despite the fact that SDP can be solved in polynomial time by IP methods, they become impractical when the number of constraints increase because of computational cost per each iteration. Rece...

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...stance from a arbitrary point to the feasible solution set of (6.34). The minimum constant LH satisfies the growth condition (6.35) is called the Hoffman constant which is well studied in [33], [27], =-=[13]-=-, [4] and [34]. That constant can be easily estimated in some cases, especially in linear system of equations. In that case, the Hoffman constant is the smallest non-zero singular value of the matrix ...

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...at σIn −Ad ∈ Sn−, and denote µ = ‖d‖ σ . (2.5) Note that the Assumption 2 implies the Slater constraint qualification condition for the feasible set of (2.4), hence S is nonempty, see [8], [33], [3], =-=[2]-=-. In Section 2 and Section 3, we will present two equivalent SDP optimization formulations of LMI and linearly convergent algorithms for solving these formulations. 3 A non-smooth SDP Optimization For...

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...ariety class of problems, see [16], [15], [30]. However, it is not easy to obtain an error bound except in linear and quadratic cases, or when the Slater constraint qualification condition holds, see =-=[3]-=-. In [33], Zhang derived error bounds for general convex conic problem under some various conditions. The error bound for Semidefinite Programming was studied by Deng and Hu in[3], Jourani and Ye in[8...

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...ndle methods [5]. In system and control theory, system identification and signal processing, Semi-definite Programmings are used in context of Linear Matrix Inequalities constraints (LMIs), see [28], =-=[31]-=-. LMIs can also be solved numerically by recent interior point methods for semi-definite programming, see [6], [26]. ∗Department of Industrial and Systems Engineering, University of Florida, Gainesvil...

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...lass of problems, see [16], [15], [30]. However, it is not easy to obtain an error bound except in linear and quadratic cases, or when the Slater constraint qualification condition holds, see [3]. In =-=[33]-=-, Zhang derived error bounds for general convex conic problem under some various conditions. The error bound for Semidefinite Programming was studied by Deng and Hu in[3], Jourani and Ye in[8]. Relate...

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...3]. In [33], Zhang derived error bounds for general convex conic problem under some various conditions. The error bound for Semidefinite Programming was studied by Deng and Hu in[3], Jourani and Ye in=-=[8]-=-. Related topics can be found in [27], [29], [14]. The paper is organized as follows. In Section 2, we introduce the problem of interest and the Slater constraint constraint qualification condition is...

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... in algorithmic convergence proofs. In particular, Luo and Tseng showed the power of error bound idea in deriving the linear convergent rate in many algorithm for variety class of problems, see [16], =-=[15]-=-, [30]. However, it is not easy to obtain an error bound except in linear and quadratic cases, or when the Slater constraint qualification condition holds, see [3]. In [33], Zhang derived error bounds...

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...ms covers both non-smooth formulation (3.6) corresponding to L = 0,M = ‖A‖ and smooth formulation (4.9) corresponding to 9 L = 2‖A‖2,M = 0, where the optimal values of both formulations are zeros. In =-=[10]-=-, Lan propose two algorithms which is uniformly optimal for solving both non-smooth and smooth convex programming problems. More interestingly, these algorithms do not require any smoothness informati...

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... Theory,... We refer to [32] for a general survey and applications of SDP. Algorithms for solving SDP have been explosively studied since a major works are made by Nesterov and Nemirovski [18], [19], =-=[20]-=-, [21], in which they showed that Interior Point (IP) methods for Linear Programming (LP) can be extended to SDP. Related topics can be found in [29], [14]. Despite the fact that SDP can be solved in ...

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...ral bundle methods [5]. In system and control theory, system identification and signal processing, Semi-definite Programmings are used in context of Linear Matrix Inequalities constraints (LMIs), see =-=[28]-=-, [31]. LMIs can also be solved numerically by recent interior point methods for semi-definite programming, see [6], [26]. ∗Department of Industrial and Systems Engineering, University of Florida, Gai...

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...arbitrary point to the feasible solution set of (6.34). The minimum constant LH satisfies the growth condition (6.35) is called the Hoffman constant which is well studied in [33], [27], [13], [4] and =-=[34]-=-. That constant can be easily estimated in some cases, especially in linear system of equations. In that case, the Hoffman constant is the smallest non-zero singular value of the matrix A. The algorit...