linear semi-infinite programming deals with the optimization of linear function als on finite-dimensional spaces under infinitely many linear constraints. For such kind of programs, a positive duality gap can occur be...
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linear semi-infinite programming deals with the optimization of linear function als on finite-dimensional spaces under infinitely many linear constraints. For such kind of programs, a positive duality gap can occur between them and their corresponding dual problems, which are linear programs posed on infinite-dimensional spaces. This paper exploits some recent existence theorems for systems of linear inequalities in order to obtain a complete classification of linear semi-infinite programming problems from the point of view of the duality gap and the viability of the discretization numerical approach. The elimination of the duality gap is also discussed. (C) 1999 Academic Press.
New algorithms for program realization of the phase method of multilinear programming were presented, and the behavior of this method at the stagnation points was studied.
New algorithms for program realization of the phase method of multilinear programming were presented, and the behavior of this method at the stagnation points was studied.
Extending the context length (i.e., the maximum supported sequence length) of LLMs is of paramount significance. To facilitate long context training of LLMs, sequence parallelism has emerged as an essential technique,...
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The dynamics of the modified canonical nonlinear programming circuit are studied and how to guarantee the stability of the network's solution. By considering the total cocontent function, the solution of the canon...
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The dynamics of the modified canonical nonlinear programming circuit are studied and how to guarantee the stability of the network's solution. By considering the total cocontent function, the solution of the canonical nonlinear programming circuit is reconciled with the problem being modeled. In addition, it is shown how the circuit can be realized using a neural network, thereby extending the results of D.W. Tank and J.J. Hopefield (ibid., ***-33, p.533-41, May 1986) to the general nonlinear programming problem.< >
This paper presents two linear cutting plane algorithms that refine existing methods for solving disjoint bilinear programs. The main idea is to avoid constructing (expensive) disjunctive facial cuts and to accelerate...
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This paper presents two linear cutting plane algorithms that refine existing methods for solving disjoint bilinear programs. The main idea is to avoid constructing (expensive) disjunctive facial cuts and to accelerate convergence through a tighter bounding scheme. These linear programming based cutting plane methods search the extreme points and cut off each one found until an exhaustive process concludes that the global minimizer is in hand. In this paper, a lower bounding step is proposed that serves to effectively fathom the remaining feasible region as not containing a global solution, thereby accelerating convergence. This is accomplished by minimizing the convex envelope of the bilinear objective over the feasible region remaining after introduction of cuts. Computational experiments demonstrate that augmenting existing methods by this simple linear programming step is surprisingly effective at identifying global solutions early by recognizing that the remaining region cannot contain an optimal solution. Numerical results for test problems from both the literature and an application area are reported.
In safety verification of hybrid systems, barrier certificates are generated by solving the verification conditions derived from non-negative representations of different types. This paper presents a new computational...
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In safety verification of hybrid systems, barrier certificates are generated by solving the verification conditions derived from non-negative representations of different types. This paper presents a new computational method, sequential linear programming projection, for directly solving the set of verification conditions represented by the Krivine-Vasilescu-Handelman's positivstellensatz. The key idea is to decompose it into two successive optimization problems that refine the desired barrier certificate and those undetermined multipliers, respectively, and solve it in an iterative scheme. The most important benefit of the proposed approach lies in that it is much more effective than the LP relaxation method in producing real barrier certificates, and possesses a much lower computational complexity than the popular sum of square relaxation methods, which is demonstrated by the theoretical analysis on complexity and the experiment on a set of examples gathered from the literature.
Due to their large variety of applications, complex optimization problems induced a great effort to develop efficient solution techniques, dealing with both continuous and discrete variables involved in nonlinear func...
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Due to their large variety of applications, complex optimization problems induced a great effort to develop efficient solution techniques, dealing with both continuous and discrete variables involved in nonlinear functions. But among the diversity of those optimization methods, the choice of the relevant technique for the treatment of a given problem keeps being a thorny issue. Within the process engineering context, batch plant design problems provide a good framework to test the performances of various optimization methods: on the one hand, two mathematical programming techniquesDICOPT++ and SBB, implemented in the GAMS environmentand on the other hand, one stochastic method, i.e., a genetic algorithm. Seven examples, showing an increasing complexity, were solved with these three techniques. The resulting comparison enables the evaluation of their efficiency in order to highlight the most appropriate method for a given problem instance. It was proved that the best performing method is SBB, even if the genetic algorithm (GA) also provides interesting solutions, in terms of quality as well as of computational time.
The branch and bound (BB) algorithm is widely used to obtain the global solution of mixed-integer linear programming (MILP) problems. On the other hand, when the traditional BB structure is directly used to solve nonc...
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The branch and bound (BB) algorithm is widely used to obtain the global solution of mixed-integer linear programming (MILP) problems. On the other hand, when the traditional BB structure is directly used to solve nonconvex mixed-integer nonlinear programming (MINLP) problems, it becomes ineffective, mainly due to the nonlinearity and nonconvexity of the feasible region of the problem. This article presents the difficulties and ineffectiveness of the direct use of the traditional BB algorithm for solving nonconvex MINLP problems and proposes the formulation of an efficient BB algorithm for solving this category of problems. The algorithm is formulated taking into account particular aspects of nonconvex MINLP problems, including (i) how to deal with the nonlinear programming (NLP) subproblems, (ii) how to detect the infeasibility of an NLP subproblem, (iii) how to treat the nonconvexity of the problem, and (iv) how to define the fathoming rules. The proposed BB algorithm is used to solve the transmission network expansion planning (TNEP) problem, a classical problem in power systems optimization, and its performance is compared with the performances of off-the-shelf optimization solvers for MINLP problems. The results obtained for four test systems, with different degrees of complexity, indicate that the proposed BB algorithm is effective for solving the TNEP problem with and without considering losses, showing equal or better performance than off-the-shelf optimization solvers.
We propose a finitely terminating primal-dual bilinear programming algorithm for the solution of the NP-hard absolute value equation (AVE): Ax -vertical bar x vertical bar = b, where A is an n x n square matrix. The a...
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We propose a finitely terminating primal-dual bilinear programming algorithm for the solution of the NP-hard absolute value equation (AVE): Ax -vertical bar x vertical bar = b, where A is an n x n square matrix. The algorithm, which makes no assumptions on AVE other than solvability, consists of a finite number of linear programs terminating at a solution of the AVE or at a stationary point of the bilinear program. The proposed algorithm was tested on 500 consecutively generated random instances of the AVE with n = 10, 50, 100, 500 and 1,000. The algorithm solved 88.6% of the test problems to an accuracy of 10(-6).
This paper presents a new concept of efficient solution for the linear vector maximization problem. Briefly, these solutions are efficient with respect to the constraints, in addition to being efficient with respect t...
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This paper presents a new concept of efficient solution for the linear vector maximization problem. Briefly, these solutions are efficient with respect to the constraints, in addition to being efficient with respect to the multiple objectives. The duality theory of linear vector maximization is developed in terms of this solution concept and then is used to formulate the problem as a linear program.
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