On nonconvex optimization with integral constraints

We consider the problem of minimizing f(y)dm with y dm=c,c fixed. The functionf is assumed to be continuous, but need not be convex. For this problem, we give necessary and sufficient conditions for the existence of solutions. We also give conditions under which uniqueness in a certain sense holds, and we show a relation which holds between the minimizers of two different problems and the corresponding values of the constraintsc.This research was supported by FINEP-Brazil, Grant Nos. 62.24-0416-00 and 4.2.82.0719-00.     

On convex optimization with linear constraints

In this paper, we propose an interior-point method for minimizing a convex function subject to linear constraints. Our method employs ideas from a previously studied method due to Fan and Nekooie in a different context. Under certain assumptions, we show that the proposed method has a fast rate of convergence. A numerical example is included to illustrate the method.     

A level set method for Laplacian eigenvalue optimization subject to geometric constraints

Computational Optimization and Applications - We consider to solve numerically the shape optimization problems of Dirichlet Laplace eigenvalues subject to volume and perimeter constraints. By...     

Farthest-point queries with geometric and combinatorial constraints

In this paper we discuss farthest-point problems in which a set or sequence S of n points in the plane is given in advance and can be preprocessed to answer various queries efficiently. First, we give a data structure that can be used to compute the point farthest from a query line segment in O(log²n) time. Our data structure needs O(nlogn) space and preprocessing time. To the best of our knowledge no solution to this problem has been suggested yet. Second, we show how to use this data structure to obtain an output-sensitive query-based algorithm for polygonal path simplification. Both results are based on a series of data structures for fundamental farthest-point queries that can be reduced to each other.     

Zero-one programming with many variables and few constraints

One of the more successful techniques for solving zero-one integer programs has been the implicit enumeration strategy first introduced by E. Balas. However, experience has shown that the efficiency of these enumerative techniques depends critically upon the bumber of variables. In this paper an algorithm is developed and computational experience provided for solving zero-one integer programs with many variables and few constraints. Sub-problems solved via implicit enumeration are generated from the linear programming relaxation and the variables in these sub-problems correspond to the fractional variables obtained in the linear program. Since the number of fractional variables in the linear program is bounded by the number of constraints in the linear program, the sub-problems will in general contain many fewer variables than the original zero-one integer program.     

On some geometric optimization problems in layered manufacturing

Efficient geometric algorithms are given for optimization problems arising in layered manufacturing, where a 3D object is built by slicing its CAD model into layers and manufacturing the layers successively. The problems considered include minimizing the stair-step error on the surfaces of the manufactured object under various formulations, minimizing the volume of the so-called support structures used, and minimizing the contact area between the supports and the manufactured object—all of which are factors that affect the speed and accuracy of the process. The stair-step minimization algorithm is valid for any polyhedron, while the support minimization algorithms are applicable only to convex polyhedra. The techniques used to obtain these results include construction and searching of certain arrangements on the sphere, 3D convex hulls, halfplane range searching, and constrained optimization.     

Monomial geometric programming with fuzzy relation equation constraints

Monomials are widely used. They are basic structural units of geometric programming. In the process of optimization, many objective functions can be denoted by monomials. We can often see them in resource allocation and structure optimization and technology management, etc. Fuzzy relation equations are important elements of fuzzy mathematics, and they have recently been widely applied in fuzzy comprehensive evaluation and cybernetics. In view of the importance of monomial functions and fuzzy relation equations, we present a fuzzy relation geometric programming model with a monomial objective function subject to the fuzzy relation equation constraints, and develop an algorithm to find an optimal solution based on the structure of the solution set of fuzzy relation equations. Two numerical examples are given to verify the developed algorithm. Our numerical results show that the algorithm is feasible and effective.     

Random linear programs with many variables and few constraints

We extend and simplify Smale's work on the expected number of pivots for a linear program with many variables and few constraints. Our analysis applies to new versions of the simplex algorithm and to new random distributions.     

Multiobjective optimization problems with equilibrium constraints

The paper is devoted to new applications of advanced tools of modern variational analysis and generalized differentiation to the study of broad classes of multiobjective optimization problems subject to equilibrium constraints in both finite-dimensional and infinite-dimensional settings. Performance criteria in multiobjective/vector optimization are defined by general preference relationships satisfying natural requirements, while equilibrium constraints are described by parameterized generalized equations/variational conditions in the sense of Robinson. Such problems are intrinsically nonsmooth and are handled in this paper via appropriate normal/coderivative/subdifferential constructions that exhibit full calculi. Most of the results obtained are new even in finite dimensions, while the case of infinite-dimensional spaces is significantly more involved requiring in addition certain “sequential normal compactness” properties of sets and mappings that are preserved under a broad spectrum of operations.     

Vector optimization problems with quasiconvex constraints

Let X be a real linear space, a convex set, Y and Z topological real linear spaces. The constrained optimization problem min _C f(x), is considered, where f : X ₀→ Y and g : X ₀→ Z are given (nonsmooth) functions, and and are closed convex cones. The weakly efficient solutions (w-minimizers) of this problem are investigated. When g obeys quasiconvex properties, first-order necessary and first-order sufficient optimality conditions in terms of Dini directional derivatives are obtained. In the special case of problems with pseudoconvex data it is shown that these conditions characterize the global w-minimizers and generalize known results from convex vector programming. The obtained results are applied to the special case of problems with finite dimensional image spaces and ordering cones the positive orthants, in particular to scalar problems with quasiconvex constraints. It is shown, that the quasiconvexity of the constraints allows to formulate the optimality conditions using the more simple single valued Dini derivatives instead of the set valued ones.      

Necessary optimality conditions for abnormal problems with geometric constraints

The abnormal minimization problem with a finite-dimensional image and geometric constraints is examined. In particular, inequality constraints are included. Second-order necessary conditions for this problem are established that strengthen previously known results.     

Zero-one integer programs with few constraints —Lower bounding theory

The zero-one integer programming problem and its special case, the multiconstraint knapsack problem frequently appear as subproblems in many combinatorial optimization problems. We present several methods for computing lower bounds on the optimal solution of the zero-one integer programming problem. They include Lagrangean, surrogate and composite relaxations. New heuristic procedures are suggested for determining good surrogate multipliers. Based on theoretical results and extensive computational testing, it is shown that for zero-one integer problems with few constraints surrogate relaxation is a viable alternative to the commonly used Lagrangean and linear programming relaxations. These results are used in a follow up paper to develop an efficient branch and bound algorithm for solving zero-one integer programming problems.     

Solving integer programs with a few important binary gub constraints

During a branch and bound search of an integer program, decisions have to be taken about which subproblem to solve next and which variable or special ordered set to branch on. Both these decisions are usually based on some sort of estimated change in the objective caused by different branching. When the next subproblem is chosen, the estimated change in the objective is often found by summing the change caused by changing all integer variables with non-integer values, as if they were independent. For special ordered sets the estimation is done for each set as a whole. The purpose of this paper is to report some results from trying to do a simultaneous estimation for all the variables in a binary gub constraint. By this, the analysed problems contain one or a few constraints saying that the sum ofn binary variables should be equal tom (<n).I am grateful to Scicon Ltd. for giving me access to the SCICONIC source code.     

On various aspects of evolutionary structural optimization for problems with stiffness constraints

An evolutionary structural optimization (ESO) method for problems with stiffness constraints which is capable of performing simultaneous shape and topology optimization has been recently presented. This paper discusses various aspects of this method such as influences of the element removal ratio, the mesh size and the element type on optimal designs.     

A primal-dual Newton-type algorithm for geometric programs with equality constraints

An augmented Lagrangian algorithm is used to find local solutions of geometric programming problems with equality constraints (GPE). The algorithm is Newton's method for unconstrained minimization. The complexity of the algorithm is defined by the number of multiplications and divisions. By analyzing the algorithm we obtain results about the influence of each parameter in the GPE problem on the complexity of an iteration. An attempt is made to estimate the number of iterations needed for convergence. In practice, certain hypotheses are tested, such as the influence of the penalty coefficient update formula, the distance of the starting point from the optimum, and the stopping criterion. For these tests, a random problem generator was constructed, many problems were run, and the results were analyzed by statistical methods.The authors are grateful to Dr. J. Moré, Argonne National Laboratory for his valuable comments.This research was partially funded by the Fund for the Advancement of Research at the Technion and by the Applied Mathematical Sciences Research Program (KC-04-02), Office of Energy Research, US Department of Energy, Contract No. W-31-109-Eng-38.     

On solving a primal geometric program by partial dual optimization

The several published methods for mapping a dual solution estimate to a primal solution estimate in posynomial geometric programming provide no criteria for deciding how much deviation from primal feasibility, or discrepancy between the primal and dual objective function values, should be permitted before the primal solution estimate is accepted by the designer. This paper presents a new and simple dual-to-primal conversion method that uses the cost coefficients to provide a sound economic criterion for determining when to accept a primal solution estimate. The primal solution estimate generated is the exact solution to a modified primal obtained from the given primal by modifying the cost coefficients, with the exponent matrix left unchanged. The method is shown to have desirable properties when coupled with a convergent dual algorithm.     

Algorithms for optimization problems with exclusion constraints

This paper proposes algorithms for minimizing a continuously differentiable functionf(x): ⁿ subject to the constraint thatx does not lie in specified bounded subsets of ⁿ . Such problems arise in a variety of applications, such as tolerance design of electronic circuits and obstacle avoidance in the selection of trajectories for robot arms. Such constraints have the form . The function is not continuously differentiable. Algorithms based on the use of generalized gradients have considerable disadvantages because of the local concavity of at points where the set {j|g ^j (x)=(x)} has more than one element. Algorithms which avoid these disadvantrages are presented, and their convergence is established.This research was sponsored in part by the National Science Foundation under Grant ECS-81-21149, the Air Force Office of Scientific Research (AFSC), United States Air Force under Contract F49620-79-C-0178, the Office of Naval Research under Grant N00014-83-K-0602, the Air Force Office of Scientific Research under Grant AFOSR-83-0361, and the Semiconductor Research Consortium under Grant SRC-82-11-008.     

ODE methods for optimization with equality constraints

In an optimization problem with equality constraints we define an accessory function that is similar but different from a normal penalty function. In the accessory function we demonstrate the need to use small values of the parameter associated with an equality constraint. Large values of the parameter create extraneous stationary points which destroy the global convergence properties of steepest descent methods. By using small values of the parameters in the accessory function, when the current point is far away from the solution and when the constraint violations are large we are led to a refined version of the established SUMT method.     

Polynomial root radius optimization with affine constraints

The root radius of a polynomial is the maximum of the moduli of its roots (zeros). We consider the following optimization problem: minimize the root radius over monic polynomials of degree n, with either real or complex coefficients, subject to k linearly independent affine constraints on the coefficients. We show that there always exists an optimal polynomial with at most \(k-1\) inactive roots, that is, roots whose moduli are strictly less than the optimal root radius. We illustrate our results using some examples arising in feedback control.     

Multiobjective optimization problem with variational inequality constraints

 We study a general multiobjective optimization problem with variational inequality, equality, inequality and abstract constraints. Fritz John type necessary optimality conditions involving Mordukhovich coderivatives are derived. They lead to Kuhn-Tucker type necessary optimality conditions under additional constraint qualifications including the calmness condition, the error bound constraint qualification, the no nonzero abnormal multiplier constraint qualification, the generalized Mangasarian-Fromovitz constraint qualification, the strong regularity constraint qualification and the linear constraint qualification. We then apply these results to the multiobjective optimization problem with complementarity constraints and the multiobjective bilevel programming problem. Received: November 2000 / Accepted: October 2001 Published online: December 19, 2002 Key Words. Multiobjective optimization – Variational inequality – Complementarity constraint – Constraint qualification – Bilevel programming problem – Preference – Utility function – Subdifferential calculus – Variational principle Research of this paper was supported by NSERC and a University of Victoria Internal Research Grant Research was supported by the National Science Foundation under grants DMS-9704203 and DMS-0102496 Mathematics Subject Classification (2000): Sub49K24, 90C29