Semi-Global Exponential Stability of Augmented Primal-Dual Gradient Dynamics for Constrained Convex Optimization

Primal-dual gradient dynamics that find saddle points of a Lagrangian have been widely employed for handling constrained optimization problems. Building on existing methods, we extend the augmented primal-dual gradient dynamics (Aug-PDGD) to incorporate general convex and nonlinear inequality constraints, and we establish its semi-global exponential stability when the objective function is strongly convex. We also provide an example of a strongly convex quadratic program of which the Aug-PDGD fails to achieve global exponential stability. Numerical simulation also suggests that the exponential convergence rate could depend on the initial distance to the KKT point

SHORT- AND LONG-RUN DEMAND AND SUBSTITUTION OF AGRICULTURAL INPUTS

Short- and long-run Hicksian and Marshallian elasticities are estimated, along with Morishima elasticities of substitution, using a restricted profit function and a series of decomposition equations. Convexity in prices and concavity in quasi-fixed factors of the restricted profit function are simultaneously imposed using Bayesian techniques. The empirical model is disaggregated in the input side, utilizes a Fuss-quadratic flexible functional form, incorporates the impact of agricultural policies, and introduces a new weather index. The methodology is applied to Illinois's agriculture, and implications for agriculture in the Corn Belt and the Northeast are briefly discussed.Demand and Price Analysis,

Research on an augmented Lagrangian penalty function algorithm for nonlinear programming

The augmented Lagrangian (ALAG) Penalty Function Algorithm for optimizing nonlinear mathematical models is discussed. The mathematical models of interest are deterministic in nature and finite dimensional optimization is assumed. A detailed review of penalty function techniques in general and the ALAG technique in particular is presented. Numerical experiments are conducted utilizing a number of nonlinear optimization problems to identify an efficient ALAG Penalty Function Technique for computer implementation

Duality and sensitivity analysis for fractional programs

"3-97-76."--handwritten on t.p. Cover title.Bibliography: p. 38-42.Supported in part by the U.S. Army Research Office (Durham) under contract no. DAHC04-732-0032 Supported in part by a Grant-in-Aid from Coca-Cola, U.S.A, administered at M.I.T. as OSP 27857by Gabriel R. Bitran and Thomas L. Magnanti

Reformulations of mathematical programming problems as linear complementarity problems

A family of complementarity problems are defined as extensions of the well known Linear Complementarity Problem (LCP). These are (i.) Second Linear Complementarity Problem (SLCP) which is an LCP extended by introducing further equality restrictions and unrestricted variables, (ii.) Minimum Linear Complementarity Problem (MLCP) which is an LCP with additional variables not required to be complementary and with a linear objective function which is to be minimized, (iii.) Second Minimum Linear Complementarity Problem (SMLCP) which is an MLCP but the nonnegative restriction on one of each pair of complementary variables is relaxed so that it is allowed to be unrestricted in value. A number of well known mathematical programming problems, namely quadratic programming (convex, nonconvex, pseudoconvex nonconvex), bilinear programming, game theory, zero-one integer programming, the fixed charge problem, absolute value programming, variable separable programming are reformulated as members of this family of four complementarity problems

Cloud-Based Centralized/Decentralized Multi-Agent Optimization with Communication Delays

We present and analyze a computational hybrid architecture for performing multi-agent optimization. The optimization problems under consideration have convex objective and constraint functions with mild smoothness conditions imposed on them. For such problems, we provide a primal-dual algorithm implemented in the hybrid architecture, which consists of a decentralized network of agents into which centralized information is occasionally injected, and we establish its convergence properties. To accomplish this, a central cloud computer aggregates global information, carries out computations of the dual variables based on this information, and then distributes the updated dual variables to the agents. The agents update their (primal) state variables and also communicate among themselves with each agent sharing and receiving state information with some number of its neighbors. Throughout, communications with the cloud are not assumed to be synchronous or instantaneous, and communication delays are explicitly accounted for in the modeling and analysis of the system. Experimental results are presented to support the theoretical developments made.Comment: 8 pages, 4 figure