We consider the questions of correction of improper convex programs, first of all, problems with inconsistent systems of constraints. Such problems often arise in the practice of mathematical simulation of specific ap...
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We consider the questions of correction of improper convex programs, first of all, problems with inconsistent systems of constraints. Such problems often arise in the practice of mathematical simulation of specific applied settings in operations research. Since improper problems are rather frequent, it is important to develop methods of their correction, i.e., methods of construction of solvable models that are close to the original problems in a certain sense. Solutions of these models are taken as generalized (approximation) solutions of the original problems. We construct the correcting problems using a variation of the right-hand sides of the constraints with respect to the minimum of a certain penalty function, which, in particular, can be taken as some norm of the vector of constraints. As a result, we obtain optimal correction methods that are modifications of the (Tikhonov) regularized method of penalty functions. Special attention is paid to the application of the exact penalty method. Convergence conditions are formulated for the proposed methods and convergence rates are established.
Question on existence of conditional approximation minimum points are considered. Is shown, that the set of points conditional approximation minimum even in case of convex functions can not cut with set approximation ...
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Question on existence of conditional approximation minimum points are considered. Is shown, that the set of points conditional approximation minimum even in case of convex functions can not cut with set approximation saddle points. However in a limit on parameter r ( r ↓ 0) these sets converge to set of points of a conditional minimum.
The principal component prsuit with reduced linear measurements (PCP_RLM) has gained great attention in applications, such as machine learning, video, and aligning multiple images. The recent research shows that stron...
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The principal component prsuit with reduced linear measurements (PCP_RLM) has gained great attention in applications, such as machine learning, video, and aligning multiple images. The recent research shows that strongly convex optimization for compressive principal component pursuit can guarantee the exact low-rank matrix recovery and sparse matrix recovery as well. In this paper, we prove that the operator of PCP_RLM satisfies restricted isometry property (RIP) with high probability. In addition, we derive the bound of parameters depending only on observed quantities based on RIP property, which will guide us how to choose suitable parameters in strongly convex programming.
A new proof of the Kuhn–Tucker necessary optimality condition is given for convex and affine inequalities. This proof differs from other existing proofs by relying only on a classical separation theorem for convex se...
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A new proof of the Kuhn–Tucker necessary optimality condition is given for convex and affine inequalities. This proof differs from other existing proofs by relying only on a classical separation theorem for convex sets. Since this statement only assumes the weaker form of the Slater constraint qualification, it contains—without unnecessary restriction—the duality theorem for linear programming as well as the optimality conditions for quadratic and convex-linear programming.
In this paper, a method is proposed for synthesizing sets of stabilizing controllers of strictly proper, delay-free, Single Input, Single Output Linear Time Invariant (LTI) plants directly from their empirical frequen...
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In this paper, a method is proposed for synthesizing sets of stabilizing controllers of strictly proper, delay-free, Single Input, Single Output Linear Time Invariant (LTI) plants directly from their empirical frequency response data and from some coarse information about them. The coarse information that is required is the following: the number of non minimum phase zeros of the plant and the frequency range beyond which the phase response of the LTI plant does not change appreciably and the amplitude response goes to zero. It is assumed that the LTI plant does not have purely imaginary zeros or poles. The method of synthesizing stabilizing controllers involves the use of generalized Hermite-Biehler theorem for rational functions for counting the roots and the use of recently developed Sum-of- Squares techniques for checking the non-negativity of a polynomial in an interval through the Markov-Lucaks theorem. The method does not require an explicit analytical model of the plant that must be stabilized or the order of the plant, rather, it only requires an empirical frequency response data of the plant. The method also allows for measurement errors in the frequency response of the plant.
A two-phase design used to sample tax records of businesses to obtain annual estimates of Canadian economic production is described. Classification information obtained from units in the first-phase sample is used dur...
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A two-phase design used to sample tax records of businesses to obtain annual estimates of Canadian economic production is described. Classification information obtained from units in the first-phase sample is used during stratification for selection of the second-phase sample. Bernoulli sampling is employed. Given multiple precision constraints, optimal allocation requires solution of a convex programming problem. An approximate method is described and compared to the optimal approach. The efficiency of the two-phase design relative to one-phase alternatives is examined.
Projection type neural network for optimization problems has advantages over other networks for fewer parameters , low searching space dimension and simple structure. In this paper, by properly constructing a Lyapunov...
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Projection type neural network for optimization problems has advantages over other networks for fewer parameters , low searching space dimension and simple structure. In this paper, by properly constructing a Lyapunov energy function, we have proven the global convergence of this network when being used to optimize a continuously differentiable convex function defined on a closed convex set. The result settles the extensive applicability of the network. Several numerical examples are given to verify the efficiency of the network.
This paper is devoted to the design of robust state feedback controllers for a class of linear uncertain systems. It is assumed that in the system and input matrices there are linear uncertain parameters, which satisf...
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This paper is devoted to the design of robust state feedback controllers for a class of linear uncertain systems. It is assumed that in the system and input matrices there are linear uncertain parameters, which satisfy no special restrictions, such as the matching condition or norm-bounded constraints. Through the matrix nonsingularity analysis, it is known that systems stabilized by a state feedback controller have robust stability bounds on the uncertain parameters, which can be computed from the structured singular value of a composite matrix. Based on this result and linear matrix inequalities, we form a convex optimization problem so that stabilizing controllers can be found to maximize the robust stability bounds.
A class of implementable methods is described for minimizing in the sense of Pareto any convex, not necessarily differentiable, vector valued function f of several variables, f(x)= (f 1 (x),...,f n (x), where each f i...
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A class of implementable methods is described for minimizing in the sense of Pareto any convex, not necessarily differentiable, vector valued function f of several variables, f(x)= (f 1 (x),...,f n (x), where each f is a convex real-valued function. The methods require only the calculation of f and one subgradient of f i , i=1,...,n, at designated points. They generalize Lemarechal’s method for minimizing convex nondifferentiable real-valued functions and Mukai’s direct descent method for solving differentiable multiobjective problems that does not require any scalarizing function. The methods are iterative and have search direction finding subproblems that are quadratic programming problems involving current subgradients of f i and an aggregate subgradient which is recursively updated as the algorithms proceed. Each algorithm yields a sequence of points with monotonically decreasing values of each objective. Any accumulation point of this sequence satisfies a necessary optimality condition for the problem considered.
A control problem that combines H 2 and H ∞ design objectives for discrete time systems is considered. Under a minimal set of assumptions, we give formulas for solving the problem of minimizing an upper bound for the...
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A control problem that combines H 2 and H ∞ design objectives for discrete time systems is considered. Under a minimal set of assumptions, we give formulas for solving the problem of minimizing an upper bound for the generalized H 2 norm of a closed loop transfer matrix subject to an H ∞ constraint on another closed loop transfer matrix; both, full information and output feedback cases are solved. The formulas are given in terms of a finite dimensional convex program over a constraint set defined by linear matrix inequalities.
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