The method of consecutive projections (MCP) is popular due to the simplicity of its implementation and efficient use of memory. The main idea of the method is that a convex set is represented as a finite or infinite i...
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The method of consecutive projections (MCP) is popular due to the simplicity of its implementation and efficient use of memory. The main idea of the method is that a convex set is represented as a finite or infinite intersection of a set of simple convex (elementary) sets. Then it is projected on the sets external to the current point. The projection on these elementary sets is very simple because they are usually semispaces. It is proved that the iterative process of consecutive projections converges, and its modifications that ensure final convergence are developed. Three subproblems are solved within the MCP at each iteration: find an elementary set for projection, determine the direction, and calculate the step length in this direction. In this paper, we make several simple propositions that make it possible to combine these three problems and accelerate the convergence of the method for solving a special class of problems called deconvolution problems.
It is well recognized the convenience of converting the linearly constrained convex optimization problems to a monotone variational ***,we have proposed a unified algorithmic framework which can guide us to construct ...
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It is well recognized the convenience of converting the linearly constrained convex optimization problems to a monotone variational ***,we have proposed a unified algorithmic framework which can guide us to construct the solution methods for solving these monotone variational *** this work,we revisit two full Jacobian decomposition of the augmented Lagrangian methods for separable convex programming which we have studied a few years *** particular,exploiting this framework,we are able to give a very clear and elementary proof of the convergence of these solution methods.
The theory of approximation algorithms has had great success with combinatorial opti-mization, where it is known that for a variety of problems, algorithms based on semidef-inite programming are optimal under the uniq...
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The theory of approximation algorithms has had great success with combinatorial opti-mization, where it is known that for a variety of problems, algorithms based on semidef-inite programming are optimal under the unique games conjecture. In contrast, the ap-proximability of most continuous optimization problems remains unresolved. In this thesis we aim to extend the theory of approximation algorithms to a wide class of continuous optimization problems captured by the injective tensor norm framework. Given an order-d tensor T, and symmetric convex sets C1,... Cd, the injective tensor norm of T is defined as sup xi Ci? T, x1 xd?, Injective tensor norm has manifestations across several branches of computer science, optimization and analysis. To list some examples, it has connections to maximum singu- lar value, max-cut, Grothendieck s inequality, non-commutative Grothendieck inequality, quantum information theory, k-XOR, refuting random constraint satisfaction problems, tensor PCA, densest-k-subgraph, and small set expansion. So a general theory of its ap- proximability promises to be of broad scope and applicability. We study various important special cases of the problem (through the lens of convex optimization and the sum of squares (SoS) hierarchy) and obtain the following results: - WeobtainthefirstNP-hardnessofapproximationresultsforhypercontractivenorms. Specifically, we prove inapproximability results for computing the p q operator norm (which is a special case of injective norm involving two convex sets) when p q and 2? [p, q]. Towards the goal of obtaining strong inapproximability results for2 q normwhen q > 2, wegiverandomlabelcover(forwhichpolynomiallevel SoS gaps are available) based hardness results for mixed norms, i. e., 2? q(? q?) for some 2 < q, q?<. - We obtain improved approximation algorithms for computing the p q operator norm when p 2 q. - We introduce the technique of weak decoupling inequalities and use it to analyze the integrality gap of the
As a powerful tool to convert nonconvex problems into convex ones, semidefinite programing (SDP) has been introduced to both cooperative and non-cooperative localization systems. In this paper, we derive the Cramé...
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The reconstruction problem of discrete tomography is studied using novel techniques from compressive sensing. Recent theoretical results of the authors enable to predict the number of measurements required for the uni...
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ISBN:
(纸本)9783319099552;9783319099545
The reconstruction problem of discrete tomography is studied using novel techniques from compressive sensing. Recent theoretical results of the authors enable to predict the number of measurements required for the unique reconstruction of a class of cosparse dense 2D and 3D signals in severely undersampled scenarios by convex programming. These results extend established l(1)-related theory based on sparsity of the signal itself to novel scenarios not covered so far, including tomographic projections of 3D solid bodies composed of few different materials. As a consequence, the large-scale optimization task based on total-variation minimization subject to tomographic projection constraints is considerably more complex than basic l(1)-programming for sparse regularization. We propose an entropic perturbation of the objective that enables to apply efficient methodologies from unconstrained optimization to the perturbed dual program. Numerical results validate the theory for large-scale recovery problems of integer-valued functions that exceed the capacity of the commercial MOSEK software.
This paper shows that the alternating direction method can be used to solve the structured inverse quadratic eigenvalue problem with symmetry, positive semi-definiteness and sparsity requirements. The results of numer...
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This paper shows that the alternating direction method can be used to solve the structured inverse quadratic eigenvalue problem with symmetry, positive semi-definiteness and sparsity requirements. The results of numerical examples show that the proposed method works well.
Electric trucks powered by hydrogen fuel cells are a promising electrified transportation technology. To compensate for the relatively slow dynamics of fuel cells, lithium-ion batteries can be incorporated into the hy...
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This paper presents a two-stage robust model predictive control (RMPC) algorithm named as IRMPC for uncertain linear integrating plants described by a state-space model with input constraints. The global convergence o...
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This paper presents a two-stage robust model predictive control (RMPC) algorithm named as IRMPC for uncertain linear integrating plants described by a state-space model with input constraints. The global convergence of the resulted closed loop system is guaranteed under mild assumption. The simulation example shows its validity and better performance than conventional Min-Max RMPC strategies.
This paper generalizes the classic resource allocation problem to the resource planning and allocation problem, in which the resource itself is a decision variable and the cost of each activity is uncertain when the r...
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This paper generalizes the classic resource allocation problem to the resource planning and allocation problem, in which the resource itself is a decision variable and the cost of each activity is uncertain when the resource is determined. The authors formulate this problem as a two-stage stochastic programming. The authors first propose an efficient algorithm for the case with finite states. Then, a sudgradient method is proposed for the general case and it is shown that the simple algorithm for the unique state case can be used to compute the subgradient of the objective function. Numerical experiments are conducted to show the effectiveness of the model.
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