A method for generating a well-distributed Pareto set in nonlinear multiobjective optimization

A method is presented for generating a well-distributed Pareto set in nonlinear multiobjective optimization. The approach shares conceptual similarity with the Physical Programming-based method, the Normal-Boundary Intersection and the Normal Constraint methods, in its systematic approach investigating the objective space in order to obtain a well-distributed Pareto set. The proposed approach is based on the generalization of the class functions which allows the orientation of the search domain to be conducted in the objective space. It is shown that the proposed modification allows the method to generate an even representation of the entire Pareto surface. The generation is performed for both convex and nonconvex Pareto frontiers. A simple algorithm has been proposed to remove local Pareto solutions. The suggested approach has been verified by several test cases, including the generation of both convex and concave Pareto frontiers.     

A row-action method for convex programming

We present a new method for minimizing a strictly convex function subject to general convex constraints. Constraints are used one at a time, no changes are made in the constraint functions (thus the row-action nature of the algorithm) and at each iteration a subproblem is solved consisting of minimization of the objective function subject to one or two linear equations. Convergence of the algorithm is established and the method is compared with other row-action algorithms for several relevant particular cases.Corresponding author. Research of this author was partially supported by CNPq grant No. 301280/86.     

Existence of general equilibrium for stochastic economy with infinite-dimensional commondity space and incomplete financial markets

This paper analyses the general equilibrium existence problem in a (finite) discretetime economy with infinite-dimensional commodity space and inComplete financial markets. It isassumed that the trading takes place in the sequence of spot markets and futures markets for sccurities payable in units of account. Unlimited short-selling in securities is allowed. The existence of such an equilibrium is proved under the following conditions: Mackey continuous,weakly convex ,strictly monotone,complete preferences and strictly positive endowments.     

A proximal-based decomposition method for convex minimization problems

This paper presents a decomposition method for solving convex minimization problems. At each iteration, the algorithm computes two proximal steps in the dual variables and one proximal step in the primal variables. We derive this algorithm from Rockafellar's proximal method of multipliers, which involves an augmented Lagrangian with an additional quadratic proximal term. The algorithm preserves the good features of the proximal method of multipliers, with the additional advantage that it leads to a decoupling of the constraints, and is thus suitable for parallel implementation. We allow for computing approximately the proximal minimization steps and we prove that under mild assumptions on the problem's data, the method is globally convergent and at a linear rate. The method is compared with alternating direction type methods and applied to the particular case of minimizing a convex function over a finite intersection of closed convex sets.Corresponding author. Partially supported by Air Force Office of Scientific Research Grant 91-0008 and National Science Foundation Grant DMS-9201297.     

A saddle-point characterization of Pareto optima

This paper provides an answer to the following basic problem of convex multi-objective optimization: Find a saddle-point condition that is both necessary and sufficient that a given point be Pareto optimal. No regularity condition is assumed for the constraints or the objectives.Research partly supported by the Natural Sciences and Engineering Research Council of Canada.Corresponding author.Contribution of this author is a part of her M.Sc. Thesis in Applied Mathematics.     

Partially finite convex programming,Part I: Quasi relative interiors and duality theory

We study convex programs that involve the minimization of a convex function over a convex subset of a topological vector space, subject to a finite number of linear inequalities. We develop the notion of the quasi relative interior of a convex set, an extension of the relative interior in finite dimensions. We use this idea in a constraint qualification for a fundamental Fenchel duality result, and then deduce duality results for these problems despite the almost invariable failure of the standard Slater condition. Part II of this work studies applications to more concrete models, whose dual problems are often finite-dimensional and computationally tractable.     

Stability in convex semi-infinite programming and rates of convergence of optimal solutions of discretized finite subproblems

We consider convex semiinfinite programming (SIP) problems with an arbitrary fixed index set T. The article analyzes the relationship between the upper and lower semicontinuity (lsc) of the optimal value function and the optimal set mapping, and the so-called Hadamard well-posedness property (allowing for more than one optimal solution). We consider the family of all functions involved in some fixed optimization problem as one element of a space of data equipped with some topology, and arbitrary perturbations are premitted as long as the perturbed problem continues to be convex semiinfinite. Since no structure is required for T, our results apply to the ordinary convex programming case. We also provide conditions, not involving any second order optimality one, guaranteeing that the distance between optimal solutions of the discretized subproblems and the optimal set of the original problem decreases by a rate which is linear with respect to the discretization mesh-size.     

Optimality conditions for non-finite valued convex composite functions

Burke (1987) has recently developed second-order necessary and sufficient conditions for convex composite optimization in the case where the convex function is finite valued. In this note we present a technique for reducing the infinite valued case to the finite valued one. We then use this technique to extend the results in Burke (1987) to the case in which the convex function may take infinite values. We conclude by comparing these results with those established by Rockafellar (1989) for the piecewise linear-quadratic case.Dedicated to the memory of Robin W. ChaneyResearch supported in part by the National Science Foundation under grants DMS-8602399 and DMS-8803206, and by the Air Force Office of Scientific Research under grant ISSA-860080.Research supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP41983.     

Convex programming for disjunctive convex optimization

Given a finite number of closed convex sets whose algebraic representation is known, we study the problem of finding the minimum of a convex function on the closure of the convex hull of the union of those sets. We derive an algebraic characterization of the feasible region in a higher-dimensional space and propose a solution procedure akin to the interior-point approach for convex programming. Received November 27, 1996 / Revised version received June 11, 1999?Published online November 9, 1999     

A simple constraint qualification in convex programming

We introduce and characterize a class of differentiable convex functions for which the Karush—Kuhn—Tucker condition is necessary for optimality. If some constraints do not belong to this class, then the characterization of optimality generally assumes an asymptotic form.We also show that for the functions that belong to this class in multi-objective optimization, Pareto solutions coincide with strong Pareto solutions,. This extends a result, well known for the linear case.Research partly supported by the National Research Council of Canada.     

Farkas-type results for fractional programming problems

Considering a constrained fractional programming problem, within the present paper we present some necessary and sufficient conditions, which ensure that the optimal objective value of the considered problem is greater than or equal to a given real constant. The desired results are obtained using the Fenchel–Lagrange duality approach applied to an optimization problem with convex or difference of convex (DC) objective functions and finitely many convex constraints. These are obtained from the initial fractional programming problem using an idea due to Dinkelbach. We also show that our general results encompass as special cases some recently obtained Farkas-type results.     

Bounded linear regularity of convex sets in Banach spaces and its applications

This paper deals with bounded linear regularity, linear regularity and the strong conical hull intersection property (CHIP) of a collection of finitely many closed convex intersecting sets in Banach spaces. It is shown that, as in finite dimensional space setting (see [6]), the standard constraint qualification implies bounded linear regularity, which in turn yields the strong conical hull intersection property, and that the collection of closed convex sets {C ₁, . . . ,C _n } is bounded linearly regular if and only if the tangent cones of {C ₁, . . . ,C _n } has the CHIP and the normal cones of {C ₁, . . . ,C _n } has the property (G)(uniformly on a neighborhood in the intersection C). As applications, we study the global error bounds for systems of linear and convex inequalities. The work of this author was partially supported by the National Natural Sciences Grant (No. 10471032) and the Excellent Young Teachers Program of MOE, P.R.C The authors thank professor K.F.Ng for his helpful discussion and the referee for their helpful suggestions on improving the first version of this paper     

The ellipsoid method and its consequences in combinatorial optimization

L. G. Khachiyan recently published a polynomial algorithm to check feasibility of a system of linear inequalities. The method is an adaptation of an algorithm proposed by Shor for non-linear optimization problems. In this paper we show that the method also yields interesting results in combinatorial optimization. Thus it yields polynomial algorithms for vertex packing in perfect graphs; for the matching and matroid intersection problems; for optimum covering of directed cuts of a digraph; for the minimum value of a submodular set function; and for other important combinatorial problems. On the negative side, it yields a proof that weighted fractional chromatic number is NP-hard. Research by the third author was supported by the Netherlands Organisation for the Advancement of Pure Research (Z.W.O.).     

Generalization of Primal—Dual Interior-Point Methods to Convex Optimization Problems in Conic Form

We generalize primal—dual interior-point methods for linear programming (LP) problems to the convex optimization problems in conic form. Previously, the most comprehensive theory of symmetric primal—dual interior-point algorithms was given by Nesterov and Todd for feasible regions expressed as the intersection of a symmetric cone with an affine subspace. In our setting, we allow an arbitrary convex cone in place of the symmetric cone. Even though some of the impressive properties attained by Nesterov—Todd algorithms are impossible in this general setting of convex optimization problems, we show that essentially all primal—dual interior-point algorithms for LP can be extended easily to the general setting. We provide three frameworks for primal—dual algorithms, each framework corresponding to a different level of sophistication in the algorithms. As the level of sophistication increases, we demand better formulations of the feasible solution sets. Our algorithms, in return, attain provably better theoretical properties. We also make a very strong connection to quasi-Newton methods by expressing the square of the symmetric primal—dual linear transformation (the so-called scaling) as a quasi-Newton update in the case of the least sophisticated framework. August 25, 1999. Final version received: March 7, 2001.     

Quantum convex support

Convex support, the mean values of a set of random variables, is central in information theory and statistics. Equally central in quantum information theory are mean values of a set of observables in a finite-dimensional C^∗-algebra A, which we call (quantum) convex support. The convex support can be viewed as a projection of the state space of A and it is a projection of a spectrahedron.Spectrahedra are increasingly investigated at least since the 1990s boom in semi-definite programming. We recall the geometry of the positive semi-definite cone and of the state space. We write a convex duality for general self-dual convex cones. This restricts to projections of state spaces and connects them to results on spectrahedra.Our main result is an analysis of the face lattice of convex support by mapping this lattice to a lattice of orthogonal projections, using natural isomorphisms. The result encodes the face lattice of the convex support into a set of projections in A and enables the integration of convex geometry with matrix calculus or algebraic techniques.     

Cone complementarity problems with finite solution sets

We introduce the notion of a complementary cone and a nondegenerate linear transformation and characterize the finiteness of the solution set of a linear complementarity problem over a closed convex cone in a finite dimensional real inner product space. In addition to the above, other geometrical properties of complementary cones have been explored.     

On the finite extension of the marginal function arising in decomposition algorithms 1

We consider the problem how a convex optimal-value function arising in primal decomposition can be finitely continued beyond its domain. By a suitable presentation of the exact penalty method an implementable continuation can be obtained which does not change the set of optimal solutions. If the problem has separability and partially linearity properties we manage to obtain a complete continuation of the optimal-value function.     

Stable monotone variational inequalities

Variational inequalities associated with monotone operators (possibly nonlinear and multivalued) and convex sets (possibly unbounded) are studied in reflexive Banach spaces. A variety of results are given which relate to a stability concept involving a natural parameter. These include characterizations useful as criteria for stable existence of solutions and also several characterizations of surjectivity. The monotone complementarity problem is covered as a special case, and the results are sharpened for linear monotone complementarity and for generalized linear programming.Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 at the University of Wisconsin - Madison and by the National Science Foundation under Grant No. DMS-8405179 at the University of Illinois at Urbana-Champaign.     

Partly convex programming and zermelo's navigation problems

Mathematical programs, that become convex programs after freezing some variables, are termed partly convex. For such programs we give saddle-point conditions that are both necessary and sufficient that a feasible point be globally optimal. The conditions require cooperation of the feasible point tested for optimality, an assumption implied by lower semicontinuity of the feasible set mapping. The characterizations are simplified if certain point-to-set mappings satisfy a sandwich condition.The tools of parametric optimization and basic point-to-set topology are used in formulating both optimality conditions and numerical methods. In particular, we solve a large class of Zermelo's navigation problems and establish global optimality of the numerical solutions.Research partly supported by NSERC of Canada.