A class of sets and a class of functions called E-convex sets and E-convex functions are introduced by relaxing the definitions of convex sets and convex functions. This kind of generalized convexity is based on the e...
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A class of sets and a class of functions called E-convex sets and E-convex functions are introduced by relaxing the definitions of convex sets and convex functions. This kind of generalized convexity is based on the effect of an operator E on the sets and domain of definition of the functions. The optimality results for E-convex programming problems are established.
This paper presents a gradient neural network model for solving convex nonlinear programming (CNP) problems. The main idea is to convert the CNP problem into an equivalent unconstrained minimization problem with objec...
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This paper presents a gradient neural network model for solving convex nonlinear programming (CNP) problems. The main idea is to convert the CNP problem into an equivalent unconstrained minimization problem with objective energy function. A gradient model is then defined directly using the derivatives of the energy function. It is also shown that the proposed neural network is stable in the sense of Lyapunov and can converge to an exact optimal solution of the original problem. It is also found that a larger scaling factor leads to a better convergence rate of the trajectory. The validity and transient behavior of the neural network are demonstrated by using various examples. (C) 2013 Elsevier Ltd. All rights reserved.
In [G.C. Feng, Z.H. Lin, B. Yu, Existence of an interior pathway to a Karush-Kuhn-Tucker point of a nonconvex programming problem, Nonlinear Anal. 32 (1998) 761-768;G.C. Feng, B. Yu, Combined homotopy interior point m...
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In [G.C. Feng, Z.H. Lin, B. Yu, Existence of an interior pathway to a Karush-Kuhn-Tucker point of a nonconvex programming problem, Nonlinear Anal. 32 (1998) 761-768;G.C. Feng, B. Yu, Combined homotopy interior point method for nonlinear programming problems, in: H. Fujita, M. Yamaguti (Eds.), Advances in Numerical Mathematics. Proceedings of the Second Japan-China Seminar on Numerical Mathematics, Lecture Notes in Numerical and Applied Analysis, vol. 14, Kinokuniya, Tokyo, 1995, pp. 9-16;Z.H. Lin, B. Yu, G.C. Feng, A combined homotopy interior point method for convex programming problem, Appl. Math. COMPLIt. 84 (1997) 193-211], a combined homotopy was constructed for solving non-convex programming and convex programming with weaker conditions, without assuming the logarithmic barrier function to be strictly convex and the solution set to be bounded. It was proven that a smooth interior path from an interior point of the feasible set to a K-K-T point of the problem exists. This shows that combined homotopy interior point methods can solve the problem that commonly used interior point methods cannot solve. However, so far, there is no result on its complexity, even for linear programming. The main difficulty is that the objective function is not monotonically decreasing on the combined homotopy path. In this paper, by taking a piecewise technique, under commonly used conditions, polynomiality of a combined homotopy interior point method is given for convex nonlinear programming. (c) 2006 Elsevier B.V. All rights reserved.
This paper gives a proof of convergence of an iterative method for maximizing a concave function subject to inequality constraints involving convex functions. The linear programming problem is an important special cas...
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This paper gives a proof of convergence of an iterative method for maximizing a concave function subject to inequality constraints involving convex functions. The linear programming problem is an important special case. The primary feature is that each iteration is very simple computationally, involving only one of the constraints. Although the paper is theoretical in nature, some numerical results are included.
Most recently, a balanced augmented Lagrangian method (ALM) has been proposed by He and Yuan for the canonical convex minimization problem with linear constraints, which advances the original ALM by balancing its subp...
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Most recently, a balanced augmented Lagrangian method (ALM) has been proposed by He and Yuan for the canonical convex minimization problem with linear constraints, which advances the original ALM by balancing its subproblems, improving its implementation and enlarging its applicable range. In this paper, we propose a dual-primal version of the newly developed balanced ALM, which updates the new iterate via a conversely dual-primal iterative order formally. The new algorithm inherits all advantages of the prototype balanced ALM, and it can be extended to more general separable convex programming problems with both linear equality and inequality constraints. The convergence analysis of the proposed method can be well conducted in the context of variational inequalities. In particular, by some application problems, we numerically validate that these balanced ALM type methods can outperform existing algorithms of the same kind significantly.
This paper provides a theoretical foundation for efficient interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled. For such problems...
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This paper provides a theoretical foundation for efficient interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled. For such problems we devise long-step and symmetric primal-dual methods. Because of the special properties of these cones and barriers, our algorithms can take steps that go typically a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier.
PROCYON is the world's first deep-space micro spacecraft launched on December 3, 2014, as a secondary payload on the launch of JAXA's Hayabusa2 spacecraft. The mission objectives include the high-speed flyby o...
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PROCYON is the world's first deep-space micro spacecraft launched on December 3, 2014, as a secondary payload on the launch of JAXA's Hayabusa2 spacecraft. The mission objectives include the high-speed flyby observation of a near-Earth asteroid, which is enabled by utilizing miniature electric propulsion and an Earth swing-by. Through the PROCYON mission design, we encountered a limitation with the zero-radius sphere-of-influence patched-conics approach used in typical mission design. The patched-conics approach could not find a trajectory toward 2000 DP107, the nominal target asteroid, which is attainable only by a distant Earth swing-by under multi-body dynamics. This limitation mainly comes from the low orbital control capabilities of small spacecraft systems. To overcome this difficulty, we propose a preliminary mission design method that quickly calculates low-thrust and gravity assist trajectories using a convex programming approach. The method enables us to search for reachable asteroids extensively under multi-body dynamical systems. We reanalyze the PROCYON mission by the proposed mission design procedure and broadly perform trade studies regarding candidate asteroids in terms of transfer costs and operational requirements. The numerical result demonstrates that the proposed method efficiently finds the nominal target that we could not find by the patched-conics approach.
In Part I of this work we derived a duality theorem for partially finite convex programs, problems for which the standard Slater condition fails almost invariably. Our result depended on a constraint qualification inv...
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In Part I of this work we derived a duality theorem for partially finite convex programs, problems for which the standard Slater condition fails almost invariably. Our result depended on a constraint qualification involving the notion of quasi relative interior. The derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function: the differentiability of various convex functions in lattices was considered at the end of Part I. In Part II we shall apply our results to a number of more concrete problems, including variants of semi-infinite linear programming, L1 approximation, constrained approximation and interpolation, spectral estimation, semi-infinite transportation problems and the generalized market area problem of Lowe and Hurter (1976). As in Part I, we shall use lattice notation extensively, but, as we illustrated there, in concrete examples lattice-theoretic ideas can be avoided, if preferred, by direct calculation.
Calibration is a common technique of data processing in survey sampling. Although convex programming would be an obvious tool for this purpose, usually other methods are used in the practice of statistical institutes....
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Calibration is a common technique of data processing in survey sampling. Although convex programming would be an obvious tool for this purpose, usually other methods are used in the practice of statistical institutes. A comparison of those methods and convex programming is reported on in this paper.
In this paper we introduce a conic optimization formulation to solve constrained convex programming, and propose a self-dual embedding model for solving the resulting conic optimization problem. The primal and dual co...
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In this paper we introduce a conic optimization formulation to solve constrained convex programming, and propose a self-dual embedding model for solving the resulting conic optimization problem. The primal and dual cones in this formulation are characterized by the original constraint functions and their corresponding conjugate functions respectively. Hence they are completely symmetric. This allows for a standard primal-dual path following approach for solving the embedded problem. Moreover, there are two immediate logarithmic barrier functions for the primal and dual cones. We show that these two logarithmic barrier functions are conjugate to each other. The explicit form of the conjugate functions are in fact not required to be known in the algorithm. An advantage of the new approach is that there is no need to assume an initial feasible solution to start with. To guarantee the polynomiality of the path-following procedure, we may apply the self-concordant barrier theory of Nesterov and Nemirovski. For this purpose, as one application, we prove that the barrier functions constructed this way are indeed self-concordant when the original constraint functions are convex and quadratic. We pose as an open question to find general conditions under which the constructed barrier functions are self-concordant.
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