This paper presents some examples of ill-behaved central paths in convex optimization. Some contain infinitely many fixed length central segments;others manifest oscillations with infinite variation. These central pat...
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This paper presents some examples of ill-behaved central paths in convex optimization. Some contain infinitely many fixed length central segments;others manifest oscillations with infinite variation. These central paths can be encountered even for infinitely differentiable data.
Augmented Lagrangian algorithms are very popular tools for solving nonlinear programming problems. At each outer iteration of these methods a simpler optimization problem is solved, for which efficient algorithms can ...
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Augmented Lagrangian algorithms are very popular tools for solving nonlinear programming problems. At each outer iteration of these methods a simpler optimization problem is solved, for which efficient algorithms can be used, especially when the problems are large. The most famous Augmented Lagrangian algorithm for minimization with inequality constraints is known as Powell-Hestenes-Rockafellar (PHR) method. The main drawback of PHR is that the objective function of the subproblems is not twice continuously differentiable. This is the main motivation for the introduction of many alternative Augmented Lagrangian methods. Most of them have interesting interpretations as proximal point methods for solving the dual problem, when the original nonlinear programming problem is convex. In this paper a numerical comparison between many of these methods is performed using all the suitable problems of the CUTE collection.
In this two-part series of papers, a new generalized minimax optimization model, termed variable programming (VP), is developed to solve dynamically a class of multi-objective optimization problems with nondecomposabl...
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In this two-part series of papers, a new generalized minimax optimization model, termed variable programming (VP), is developed to solve dynamically a class of multi-objective optimization problems with nondecomposable structure. It is demonstrated that such type of problems is more general than existing optimization models. In this part, the VP model is proposed first, and the relationship between variable programming and the general constrained nonlinear programming is established. To illustrate its practicality, problems on investment and the low-side-lobe conformal antenna array pattern synthesis to which VP can be appropriately applied are discussed for substantiation. Then, theoretical underpinnings of the VP problems are established. Difficulties in dealing with the VP problems are discussed. With some mild assumptions, the necessary conditions for the unconstrained VP problems with arbitrary and specific activated feasible sets are derived respectively. The necessary conditions for the corresponding constrained VP problems with the mild hypotheses are also examined. Whilst discussion in this part is concentrated on the formulation of the VP model and its theoretical underpinnings, construction of solution algorithms is discussed in Part II.
Dynamic voltage scaling and adaptive body biasing have been shown to reduce dynamic and leakage power consumption effectively. The authors report an optimal solution to the combined supply voltage and body bias select...
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Dynamic voltage scaling and adaptive body biasing have been shown to reduce dynamic and leakage power consumption effectively. The authors report an optimal solution to the combined supply voltage and body bias selection problem for multiprocessor systems with imposed time constraints, explicitly taking into account the transition overheads implied by changing voltage levels, and considering both energy and time overheads. They investigate continuous voltage scaling as well as its discrete counterpart, and strongly prove NP-hardness in the discrete case. Furthermore, the continuous voltage scaling problem is formulated and solved using nonlinear programming with polynomial time complexity, while for the discrete problem they use mixed integer linear programming. Extensive experiments, conducted on several benchmarks and a real-life example, are used to validate the approaches.
This paper deals with optimization of functions that depend on (large-scale) data via a linear transformation of rank two. An algorithm is presented which-under mild assumptions-finds the global solution with polynomi...
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This paper deals with optimization of functions that depend on (large-scale) data via a linear transformation of rank two. An algorithm is presented which-under mild assumptions-finds the global solution with polynomial-time complexity in the worst case, provided the critical points of the objective can be controlled with the same effort. Moreover, for important subclasses of objectives including the linear-fractional, and the quadratic case, we arrive at a linear-time algorithm. For both cases, small simulation studies are provided to illustrate the average case runtime behaviour. As a possible application, sensitivity of cost assessment in communication networks is addressed where the problems may have tens of thousands of variables. (C) 2003 Elsevier B.V. All rights reserved.
Maximum Boolean satisfiability (max-SAT) is the optimization counterpart of Boolean satisfiability (SAT), in which a variable assignment is sought to satisfy the maximum number of clauses in a Boolean formula. A branc...
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Maximum Boolean satisfiability (max-SAT) is the optimization counterpart of Boolean satisfiability (SAT), in which a variable assignment is sought to satisfy the maximum number of clauses in a Boolean formula. A branch and bound algorithm based on the Davis-Putnain-Logemann-Loveland procedure (DPLL) is one of the most competitive exact algorithms for solving max-SAT. In this paper, we propose and investigate a number of strategies for max-SAT. The first strategy is a set of unit propagation or unit resolution rules for max-SAT. We summarize three existing unit propagation rules and propose a new one based on a nonlinear programming formulation of max-SAT. The second strategy is an effective lower bound based on linear programming (LP). We show that the LP lower bound can be made effective as the number of clauses increases. The third strategy consists of a binary-clause first rule and a dynamic-weighting variable ordering rule, which are motivated by a thorough analysis of two existing well-known variable orderings. Based on the analysis of these strategies, we develop an exact solver for both max-SAT and weighted max-SAT. Our experimental results on random problem instances and many instances from the max-SAT libraries show that our new solver outperforms most of the existing exact max-SAT solvers, with orders of magnitude of improvement in many cases. (c) 2005 Elsevier B.V. All rights reserved.
The low-rank semidefinite programming problem LRSDPr is a restriction of the semidefinite programming problem SDP in which a bound r is imposed on the rank of X, and it is well known that LRSDPr is equivalent to SDP i...
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The low-rank semidefinite programming problem LRSDPr is a restriction of the semidefinite programming problem SDP in which a bound r is imposed on the rank of X, and it is well known that LRSDPr is equivalent to SDP if r is not too small. In this paper, we classify the local minima of LRSDPr and prove the optimal convergence of a slight variant of the successful, yet experimental, algorithm of Burer and Monteiro [5], which handles LRSDPr via the nonconvex change of variables X = RRT. In addition, for particular problem classes, we describe a practical technique for obtaining lower bounds on the optimal solution value during the execution of the algorithm. Computational results are presented on a set of combinatorial optimization relaxations, including some of the largest quadratic assignment SDPs solved to date.
A new method for constructing constraints and optimizing sensory attributes of a frankfurter-type sausage is presented. Small batches of 87 sausages with different biochemical composition and meat content were produce...
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A new method for constructing constraints and optimizing sensory attributes of a frankfurter-type sausage is presented. Small batches of 87 sausages with different biochemical composition and meat content were produced. The colour and firmness were instrumentally measured and modelled as mathematical functions of biochemical composition and muscle content. Ten varieties was selected based on these colour and firmness measurements and subsequently produced in larger quantities and served to consumers in order to identify acceptability limits for colour and firmness parameters of sausage. These limits together with the mathematical functions were used as quality constraints in an overall nonlinear optimization model, to find least-cost formulas that provide sausages with acceptable sensory quality to the consumer. Consumer acceptability limits of lightness defined as 60% purchase criterion were found to be between 62 and 68 (L*-CIE colour scale), based on a selection of five sausage samples. For firmness this was more complicated because of a high content of connective tissue in some of the sausage samples. It was possible to achieve the lightness and firmness preferred by the consumers without satisfying the legal restrictions for biochemical composition. In addition, the formula costs less when the solution fulfils consumer restrictions instead of legal restrictions. (c) 2004 Swiss Society of Food Science and Technology. Published by Elsevier Ltd. All rights reserved.
A mathematical program is proposed for the highly nonlinear problem involving frictional contact. A program-pattern using the fast multipole boundary element method (FM- BEM) is given for 3-D elastic contact with fric...
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A mathematical program is proposed for the highly nonlinear problem involving frictional contact. A program-pattern using the fast multipole boundary element method (FM- BEM) is given for 3-D elastic contact with friction to replace the Monte Carlo method. A new optimized generalized minimal residual (GMRES) algorithm is presented. Numerical examples demonstrate the validity of the program-pattern optimization model for node-to-surface contact with friction. The GMRES algorithm greatly improves the computational efciency.
We develop an algorithm to compute optimal policies for Markov decision processes subject to constraints that result from some observability restrictions on the process. We assume that the state of the Markov process ...
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We develop an algorithm to compute optimal policies for Markov decision processes subject to constraints that result from some observability restrictions on the process. We assume that the state of the Markov process is unobservable. There is an observable process related to the unobservable state. So, we want to find a decision rule depending only on this observable process. The objective is to minimize the expected average cost over an infinite horizon. We also analyze the possibility of performing observations in more detail to obtain improved policies.
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