Application of general semi-infinite Programming to Lapidary Cutting Problems

We consider a volume maximization problem arising in gemstone cutting industry. The problem is formulated as a general semi-infinite program (GSIP) and solved using an interiorpoint method developed by Stein. It is shown, that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling. Clustering techniques are used to reduce the number of container constraints, which is necessary to make the subproblems practically tractable. An iterative process consisting of GSIP optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem. Some numerical results based on realworld data are also presented

Application of general semi-infinite Programming to Lapidary Cutting Problems

We consider a volume maximization problem arising in gemstone cutting industry. The problem is formulated as a general semi-infinite program (GSIP) and solved using an interiorpoint method developed by Stein. It is shown, that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling. Clustering techniques are used to reduce the number of container constraints, which is necessary to make the subproblems practically tractable. An iterative process consisting of GSIP optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem. Some numerical results based on realworld data are also presented

Application of general semi-infinite programming to lapidary cutting problems

We consider a volume maximization problem arising in gemstone cutting industry. The problem is formulated as a general semi-infinite program (GSIP) and solved using an interior-point method developed by Stein [O. Stein, Bi-level Strategies in Semi-infinite Programming, Kluwer Academic Publishers, Boston, 2003]. It is shown, that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling. Clustering techniques are used to reduce the number of container constraints, which is necessary to make the subproblems practically tractable. An iterative process consisting of GSIP optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem. Some numerical results based on real-world data are also presented.

New development of the inclusive-cone-based method for linear optimization

The purpose of this dissertation is to present a simple method for linear optimization including linear programming and linear semi-infinite programming, which is termed “the inclusive-cone-based method”. Using the inclusive cone as an analytic tool, theoretical aspects of linear programming are investigated. Sensitivity analysis in linear programming is examined from the perspective of an inclusive cone. The relationship of inclusiveness between correlated linear programming problems is also studied. New inclusive-cone-based ladder algorithms are proposed to solve linear programming problems in inequality form. Numerical experiments are implemented to show effectiveness and efficiency of the new linear programming ladder algorithms. To start the ladder method for linear programming problems, a single artificial constraint technique is introduced to find an initial ladder. Further, in the context of a new category of linear programming problems, an inclusive-cone-based solvability criterion is established to distinguish that a linear programming problem is inclusive-feasible (i.e., optimal), noninclusive-feasible (i.e., unbounded), inclusive-infeasible or noninclusive-infeasible. The inclusive-cone-based method for linear programming is also generalized to linear semi-infinite programming. An optimality result, based upon the concept of the generalized base point, is established. With this optimality result as a theoretical foundation, a ladder algorithm for solving linear semi-infinite programming problems is developed. The new algorithm has several features: at each iteration it only deals with a small fraction of constraints; at each iteration it selects a constraint most violated along a “parameterized centreline”, by solving a one-dimensional global optimization problem using the efficient bridging algorithm; at each iteration the selection of the incoming constraint has a great degree of freedom, which is controlled by a parameter arising in the global optimization problem; it can detect infeasibility and unboundedness after a finite number of iterations; it obviates extra work for feasibility verification as it handles feasibility and optimality simultaneously. A simple convergent result is presented. Numerical behaviour of the algorithm is examined on several test problems

Global optimization algorithms for semi-infinite and generalized semi-infinite programs

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2008.Includes bibliographical references (p. 235-249).The goals of this thesis are the development of global optimization algorithms for semi-infinite and generalized semi-infinite programs and the application of these algorithms to kinetic model reduction. The outstanding issue with semi-infinite programming (SIP) was a methodology that could provide a certificate of global optimality on finite termination for SIP with nonconvex functions participating. We have developed the first methodology that can generate guaranteed feasible points for SIP and provide e-global optimality on finite termination. The algorithm has been implemented in a branch-and-bound (B&B) framework and uses discretization coupled with convexification for the lower bounding problem and the interval constrained reformulation for the upper bounding problem. Within the framework of SIP we have also proposed a number of feasible-point methods that all rely on the same basic principle; the relaxation of the lower-level problem causes a restriction of the outer problem and vice versa. All these methodologies were tested using the Watson test set. It was concluded that the concave overestimation of the SIP constraint using McCormcick relaxations and a KKT treatment of the resulting expression is the most computationally expensive method but provides tighter bounds than the interval constrained reformulation or a concave overestimator of the SIP constraint followed by linearization. All methods can work very efficiently for small problems (1-3 parameters) but suffer from the drawback that in order to converge to the global solution value the parameter set needs to subdivided. Therefore, for problems with more than 4 parameters, intractable subproblems arise very high in the B&B tree and render global solution of the whole problem infeasible.(cont.) The second contribution of the thesis was the development of the first finite procedure that generates guaranteed feasible points and a certificate of e-global optimality for generalized semi-infinite programs (GSIP) with nonconvex functions participating. The algorithm employs interval extensions on the lower-level inequality constraints and then uses discretization and the interval constrained reformulation for the lower and upper bounding subproblems, respectively. We have demonstrated that our method can handle the irregular behavior of GSIP, such as the non-closedness of the feasible set, the existence of re-entrant corner points, the infimum not being attained and above all, problems with nonconvex functions participating. Finally, we have proposed an extensive test set consisting of both literature an original examples. Similar to the case of SIP, to guarantee e-convergence the parameter set needs to be subdivided and therefore, only small examples (1-3 parameters) can be handled in this framework in reasonable computational times (at present). The final contribution of the thesis was the development of techniques to provide optimal ranges of valid reduction between full and reduced kinetic models. First of all, we demonstrated that kinetic model reduction is a design centering problem and explored alternative optimization formulations such as SIP, GSIP and bilevel programming. Secondly, we showed that our SIP and GSIP techniques are probably not capable of handling large-scale systems, even if kinetic model reduction has a very special structure, because of the need for subdivision which leads to an explosion in the number of constraints. Finally, we propose alternative ways of estimating feasible regions of valid reduction using interval theory, critical points and line minimization.by Panayiotis Lemonidis.Ph.D