Global Optimality Conditions Research Articles

Global optimality conditions and exact penalization

This paper addresses the nonconvex optimization problem with the cost function and the inequality constraints given by d.c. functions. The original problem is reduced to a problem without inequality constraints by means of the exact penalization techniques. Furthermore, the penalized problem is presented as a d.c. minimization problem. For the latter problem we develop the global optimality conditions (GOCs) which reduce the nonconvex optimization problem to a family of convex (linearized with respect to the basic nonconvexity) problems. In addition,the GOCs are related to the KKT theorem for the original problem. Besides, the GOCs possess the so-called constructive (algorithmic) property which, if the GOCs are violated, implies the construction of a feasible point that is better (in the sense of the original problem) than the one in question. The effectiveness of the GOCs is demonstrated by examples.

Strong Stationarity Conditions for Optimal Control of Hybrid Systems

We present necessary and sufficient optimality conditions for finite-time optimal control problems for a class of hybrid systems described by linear complementarity models. Although these optimal control problems are difficult in general due to the presence of complementarity constraints, we provide a set of structural assumptions ensuring that the tangent cone of the constraints possesses geometric regularity properties. These imply that the classical Karush–Kuhn–Tucker conditions of nonlinear programming theory are both necessary and sufficient for local optimality, which is not the case for general mathematical programs with complementarity constraints. We also present sufficient conditions for global optimality. We proceed to show that the dynamics of every continuous piecewise affine (PWA) system can be written as the optimizer of a mathematical program, which results in a linear complementarity model satisfying our structural assumptions. Hence, our stationarity results apply to a large class of hybrid systems with PWA dynamics. We present simulation results showing the substantial benefits possible from using a nonlinear programming approach to the optimal control problem with complementarity constraints instead of a more traditional mixed-integer formulation.

Comparing large-sample maximum Sharpe ratios and incremental variable testing

Most existing results on the distribution of the maximum Sharpe ratio depend on the assumption of multivariate normal return distributions. We use recent results from the literature to provide an analytical representation of the distribution of the difference between two maximum Sharpe ratios for much less restrictive distributional assumptions, both with and without short sales. Knowing the distribution of the difference enables us to test ex ante whether or not the inclusion of additional variables leads to a significant improvement in the maximum Sharpe ratio. In addition, we characterize the optimal long-only solution and provide conditions for global optimality.

Global Optimality Conditions in Nonconvex Optimization

In this paper, we address the nonconvex optimization problem, with the goal function and the inequality constraints given by the functions represented by the difference of convex functions. The effectiveness of the classical Lagrange function and the max-merit function is being investigated as the merit functions of the original problem. In addition to the classical apparatus of optimization theory, we apply the new global optimality conditions for the auxiliary problems related to the Lagrange and max-merit functions.

Solving Multiextremal Problems by Using Recurrent Neural Networks.

In this paper, a neural network model for solving a class of multiextremal smooth nonconvex constrained optimization problems is proposed. Neural network is designed in such a way that its equilibrium points coincide with the local and global optimal solutions of the corresponding optimization problem. Based on the suitable underestimators for the Lagrangian of the problem, one give geometric criteria for an equilibrium point to be a global minimizer of multiextremal constrained optimization problem with or without bounds on the variables. Both necessary and sufficient global optimality conditions for a class of multiextremal constrained optimization problems are presented to determine a global optimal solution. By study of the resulting dynamic system, it is shown that under given assumptions, steady states of the dynamic system are stable and trajectories of the proposed model converge to the local and global optimal solutions of the problem. Numerical results are given and related graphs are depicted to illustrate the global convergence and performance of the solver for multiextremal constrained optimization problems.

Parametric Dual Maximization for Non-Convex Learning Problems

We consider a class of non-convex learning problems that can be formulated as jointly optimizing regularized hinge loss and a set of auxiliary variables. Such problems encompass but are not limited to various versions of semi-supervised learning,learning with hidden structures, robust learning, etc. Existing methods either suffer from local minima or have to invoke anon-scalable combinatorial search. In this paper, we propose a novel learning procedure, namely Parametric Dual Maximization(PDM), that can approach global optimality efficiently with user specified approximation levels. The building blocks of PDM are two new results: (1) The equivalent convex maximization reformulation derived by parametric analysis.(2) The improvement of local solutions based on a necessary and sufficient condition for global optimality. Experimental results on two representative applications demonstrate the effectiveness of PDM compared to other approaches.

Valorisation of tomato wastes for development of nutrient-rich antioxidant ingredients: A sustainable approach towards the needs of the today's society

Abstract Nutrient-rich antioxidant ingredients were produced from tomato fruit wastes using a microwave-assisted extraction (MAE) process. Different conditions of extraction time ( t ), temperature ( T ), ethanol concentration ( Et ) and solid/liquid ratio (S/L) were combined in a circumscribed central composite design and optimized by response surface methodology. The model was statistically validated and used for prediction in the experimental range. Under the global optimal MAE conditions ( t = 20 min, T = 180 °C, Et = 47.4% and S/L = 45 g/L), it was possible to obtain an extraction yield of 75.5% and ingredients with high levels of sugars, proteins, phenolics, and flavonoids, and interesting antioxidant properties measured via ABTS + scavenging activity and oxidative haemolysis inhibition assay (OxHLIA). The antioxidant capacity of the extracts was lower compared to the one of commercial food additives. However, the sustainably developed ingredients may be used in the fortification and functionalisation of food, as well as for incorporation in feed products. Industrial relevance This study addresses current needs of the agri-food sector, namely the recycling of plant wastes and production of valuable extracts for the food/feed industry. A MAE process was developed and optimized to maximize the recovery of nutrients and antioxidants from tomato fruit wastes. The optimum processing conditions established in this study allowed a high extraction yield and reduced solvent consumption. MAE can be considered as a sustainable alternative to conventional extraction methods. These findings will contribute to promote a more sustainable bioeconomy in the agro-food sector.

Some remarks on duality and optimality of a class of constrained convex quadratic minimization problems

In this paper the duality and optimality of a class of constrained convex quadratic optimization problems have been studied. Furthermore, the global optimality condition of a class of interval quadratic minimization problems has also been discussed.

Multi-parameter Optimization of a Thermoelectric Power Generator and Its Working Conditions

The global optimal working conditions and optimal couple design for thermoelectric (TE) generators with realistic thermal coupling between the heat reservoirs and the TE couple were studied in the current work. The heat fluxes enforced by the heat reservoirs at the hot and the cold junctions of the TE couple were used in combination with parameter normalization to obtain a single cubic algebraic equation relating the temperature differences between the TE couple junctions and between the heat reservoirs, through the electric load resistance ratio, the reservoir thermal conductance ratio, the reservoir thermal conductance to the TE couple thermal conductance ratio, the Thomson to Seebeck coefficient ratio, and the figure of merit (Z) of the material based on the linear TE transport equations and their solutions. A broad reservoir thermal conductance ranging between 0.01 W/K and 100 W/K and TE element length ranging from 10-7 m to 10-3 m were explored to find the global optimal systems. The global optimal parameters related to the working conditions, i.e., reservoir thermal conductance ratio and electric load resistance ratio, and the optimal design parameter related to the TE couple were determined for a given TE material. These results demonstrated that the internal and external electric resistance, the thermal resistance between the reservoirs, the thermal resistance between the reservoir and the TE couple, and the optimal thermoelement length have to be well coordinated to obtain optimal power production.

Global Optimization on an Interval

This paper provides expressions for solutions of a one-dimensional global optimization problem using an adjoint variable which represents the available one-sided improvements up to the interval “horizon.” Interpreting the problem in terms of optimal stopping or optimal starting, the solution characterization yields two-point boundary problems as in dynamic optimization. Results also include a procedure for computing the adjoint variable, as well as necessary and sufficient global optimality conditions.

Global Optimal Trajectory in Chaos and NP-Hardness

This paper presents an unconventional theory and method for solving general nonlinear dynamical systems. Instead of the direct iterative methods, the discretized nonlinear system is first formulated as a global optimization problem via the least squares method. A newly developed canonical duality theory shows that this nonconvex minimization problem can be solved deterministically in polynomial time if a global optimality condition is satisfied. The so-called pseudo-chaos produced by linear iterative methods are mainly due to the intrinsic numerical error accumulations. Otherwise, the global optimization problem could be NP-hard and the nonlinear system can be really chaotic. A conjecture is proposed, which reveals the connection between chaos in nonlinear dynamics and NP-hardness in computer science. The methodology and the conjecture are verified by applications to the well-known logistic equation, a forced memristive circuit and the Lorenz system. Computational results show that the canonical duality theory can be used to identify chaotic systems and to obtain realistic global optimal solutions in nonlinear dynamical systems. The method and results presented in this paper should bring some new insights into nonlinear dynamical systems and NP-hardness in computational complexity theory.

Optimization of microwave-assisted extraction of ergosterol from Agaricus bisporus L. by-products using response surface methodology

This work intends to valorize by-products of the industrial processing of mushrooms to obtain ergosterol as a value-added compound. Agaricus bisporus L. is the world's most consumed mushroom and one of the richest sources of ergosterol. Microwave-assisted extraction was used to replace conventional techniques that are time-consuming and need large amounts of solvent. Time (3–20min), temperature (60–210°C) and solid-liquid ratio (1–20g/L) were found the relevant variables to analyze the extraction process. To maximize the ergosterol extraction yield, response surface methodology was used to optimize the process. The global optimal extraction conditions were determined and comprise: 19.4±2.9min, 132.8±12.4°C and 1.6±0.5g/L, yielding 556.1±26.2mg of ergosterol per 100g of mushroom by-products. In the MAE optimal conditions, it was possible to obtain ergosterol in a similar value to the one obtained in other works when using the Soxhlet extraction method with a significant decrease in the time of extraction. The results show the potential of using the by-products of an agroindustry, mushrooms processing industry, as productive sources of ergosterol.

Solving Malfatti's high dimensional problem by global optimization

We generalize Malfatti's problem which dates back to 200 years ago as a global optimization problem in a high dimensional space. The problem has been formulated as the convex maximization problem over a nonconvex set. Global optimality condition by Strekalovsky [11] has been applied to this problem. For solving numerically Malfatti's problem, we propose the algorithm in [3] which converges globally. Some computational results are provided.

On global search in nonconvex optimal control problems

Optimal control problems with a nonconvex quadratic functional of Lagrange are considered. On the base of global optimality conditions we develop a global search algorithm, one of the principal module of which is represented by special local search method. The results of computational testing presented.

Global optimization approach to Malfatti’s problem

We examine Malfatti's problem which dates back to 200 years ago from the view point of global optimization. The problem has been formulated as the convex maximization problem over a nonconvex set. Global optimality condition by Strekalovsky (Sov Math Dokl 292(5):1062---1066, 1987) has been applied to this problem. For solving numerically Malfatti's problem, we propose the algorithm in Enkhbat (J Glob Optim 8:379---391, 1996) which converges globally. Some computational results are provided.

Active set solver for min-max robust control with state and input constraints

This paper proposes an online active set strategy for computing the dynamic programming solution to a min-max robust optimal control problem with quadratic H1 stage cost for linear systems with linear state and input constraints in the presence of bounded disturbances. The solver determines the optimal active constraint set for a given plant state using an iterative procedure which computes the optimal sequence of feedback laws for a candidate active set and updates the active set by performing a line search in state space. The computational complexity of each iteration depends linearly on the length of the prediction horizon. The main contribution of the paper is its treatment of degeneracy caused by linearly dependent state and input constraints and its efficient handling is a crucial step in formulating the active set algorithm. The proposed approach ensures the continuity of optimal control laws along the line-of-search, thus enabling an efficient solution method based on homotopy. Conditions for global optimality are given and the convergence of the active set solver is established using the geometric properties of an associated multi-parametric programming problem. A receding horizon control strategy is proposed, which ensures a specified l2-gain from the disturbance input to the state and control inputs in the presence of linearly dependent constraints

Trust region subproblem with an additional linear inequality constraint

This paper studies an extended trust region subproblem (eTRS) in which the trust region intersects the unit ball with a single linear inequality constraint. We present an efficient algorithm to solve the problem using a diagonalization scheme that requires solving a simple convex minimization problem. Attainment of the global optimality conditions is discussed. Our preliminary numerical experiments on several randomly generated test problems show that, the new approach is much faster in finding the global optimal solution than the known semidefinite relaxation approach, especially when solving large scale problems.

Global optimality conditions for nonconvex minimization problems with quadratic constraints

In this paper, some global optimality conditions for nonconvex minimization problems subject to quadratic inequality constraints are presented. Then some sufficient and necessary global optimality conditions for nonlinear programming problems with box constraints are derived. We also establish a sufficient global optimality condition for a nonconvex quadratic minimization problem with box constraints, which is expressed in a simple way in terms of the problem’s data. In addition, a sufficient and necessary global optimality condition for a class of nonconvex quadratic programming problems with box constraints is discussed. We also present some numerical examples to illustrate the significance of our optimality conditions.

GLOBAL OPTIMIZATION APPROACH TO UTILITY MAXIMIZATION PROBLEM

We consider the utility maximization problem for oligopsonistic market which is nonconvex optimization problem. Unlike the utility maximization for competitive market, the problem belongs to a class of global optimization. The purpose of this paper is to develop a theory and method for the above problem. We derive a new global optimality condition for our problem and based on this we propose a method which converges globally. Some test problems are examined. AMS Subject Classification: 49K30, 90C90, 90C26, 91B42

Global optimality conditions for cubic minimization problems with cubic constraints

In this paper, we present global optimality conditions for cubic minimization involving cubic constraints and box or bivalent constraints, where the cubic objective function and cubic constraints contain no cross terms. By utilizing quadratic underestimators, we first derive sufficient global optimality conditions for a global minimizer of cubic minimization problems with cubic inequality and box constraints. Then we establish them for cubic minimization with cubic inequality and bivalent constraints. Finally, we establish sufficient and necessary global optimality condition for cubic minimization with cubic equality and binary constraints.