Abstract: An embodiment relates to a multi-objective optimization design supporting technique to reduces the computational complexity of QE/CAD. When the input logical expression generated by a logical-expression-with-qualifier generation unit is satisfied in regard to the sample point included in a certain piece of cell information for each value of the same design parameter, a first cell processing unit does not evaluate the input logical expression on the cell information including other sample points having a value equal to or smaller than the value of a predetermined design variable (for example, a design variable indicating a yield) corresponding to the sample point above, but selects it as the cell information for an output of a logical expression without a qualifier.

Abstract: A design support apparatus includes a parameter set generation unit configured to obtain a plurality of types of parameters and sequentially generates parameter sets while sequentially changing each parameter, a design object shape data generation unit configured to generate design object shape data based on the parameter set and initial shape data representing an initial shape of the design object shape, a geometric penalty function value calculation unit configured to calculate a geometric penalty function value indicating suitability of geometric characteristics of the design object shape based on the design object shape data, an objective function calculation control unit configured to determine whether or not the parameter set is used to calculate an objective function based on the geometric penalty function value and an optimal value of the objective function, and an objective function calculation unit configured to calculate the objective function based on the parameter set.

Abstract: A design support apparatus includes: a logical expression substitution unit to substitute a part of the logical expression, which includes a function expression of the design variables and a quantifier attached to the design variable, with a substitution variable; a quantifier elimination unit to generate a relational expression including the substitution variable and design variables without the quantifier by eliminating the design variable to which the quantifier is attached from the logical expression; a sampling point generation unit to generate a plurality of sampling points corresponding to the design variables and the substitution variable included in the relational expression; a possible range computation unit to compute, for each of the sampling points, a possible range that the relational expression may take, by calculating values of remaining design variables included in the relational expression based on the relational expression; and a possible range display unit to display the possible range.

Abstract: A model parameter determination program stored in a storage medium used to direct a computer to efficiently and effectively determine a parameter of a model in a simulation, and includes a step of receiving input of data of a variable in a model of a system to be analyzed; a step of generating a constraint expression relating to a parameter of a model based on the received data and the model of the system; and a step of obtaining presence/absence of a solution and/or a possible range of a feasible parameter by applying a quantifier elimination algorithm to the constraint expression.

Abstract: A design support apparatus for determining a plurality of objective functions for modeling an object having a plurality of elements, each of the elements providing variable geometrical parameters, the design support apparatus includes a memory for storing the variable geometrical parameters and a processor for executing a process including: determining boundary information associated with specified geometrical parameters of the elements which indicate a state of contact between the elements, dividing the variable geometrical parameters into a plurality of groups on the basis of the boundary information, and determining the plurality of objective functions for each of the groups by using the variable geometrical parameters.

Abstract: A program causes a computer to perform a step 1 of reading a model from a storage apparatus; a step 2 of computing difference equations by replacing variables with numeric values of a time-series that are stored in the storage apparatus; a step 3 of determining ranges of parameters of the model using a quantifier elimination method and a step 4 of verifying numbers of significant figures that show an accuracy of a computation by the determined ranges of parameters of the model.

Abstract: An apparatus receives input of sample sets, each including a set of values of design parameters and a set of values of objective functions; calculates objective function approximating equations; and selects, as initial candidates for an optimal design parameter set, some sets of values of design parameters corresponding to non-dominated solutions. The apparatus calculates one or more interpolating design parameter sets interpolating between two adjacent components in the candidates; and approximates values of the objective functions for each interpolating design parameter set. The apparatus selects an optimal interpolating design parameter set corresponding to a non-dominated solution in the cost evaluation for a pair of objective functions; and integrates it into the candidates. The apparatus repeats processes on the new candidates while determining the parameter distance between components of the new candidates.

Abstract: A system displays an area which a desired objection function value of a plurality of objective functions as a possible area in objective space corresponding to the objective function on the basis of each of the plurality of objective function value sets calculated for a plurality of design parameter sample sets; calculates a design parameter set in design space corresponding to the neighborhood area of a position in the objective space based on the position specification in relation to position specification by a user in the possible area of the objective space; and calculates and displays a representative design shape corresponding to the calculated design parameter set.

Abstract: A multi-objective optical design improvement support device calculates a logical expression indicating a logical relationship among arbitrary two or three objective functions of a plurality of mathematically approximated objective functions and displays a possibility area in arbitrary objective space according to it. When a designer is not satisfied with an optimal Pareto solution, it copies a sample point group out of the initial constraints of a design parameter set in the objective space, displays its result and presents an improvement solution to the designer. When the designer finds more optimal solution than the optimal Pareto solution among the displayed improvement solutions and gives instruction, it calculates a sample point in design parameter space, corresponding to the optimal improvement solution, overlaps it with a constraint range and displays it as improvement knowledge and information.

Abstract: A logical expression indicating a logical relation between arbitrary two or three objective functions, of a plurality of mathematically approximated objective functions is computed. A feasible region/sensitivity information display unit displays the feasible region in arbitrary objective space according to it. An inverse image computation unit computes a point or area in design space corresponding to arbitrary design parameters in relation to a point or area specified by a user in the feasible region of the objective space. A feasible region/sensitivity information display unit displays the distribution state of the corresponding point or area as sensitivity information in relation to the specified point or area in the feasible region.

Abstract: A unit 101 calculates a plurality of sets of objective functions of sample sets of input parameters. A unit 102 approximates the objective functions using a polynomial on the basis of the calculation result of the unit 101. A unit 103 calculates a logical expression indicating a logical relation between arbitrary two or three objective functions, of the plurality of mathematically approximated objective functions as an inter-objective function logical expression by a QE method. A unit 104 displays areas that the values of the objective functions can take as usable areas according to the inter-objective function logical expression. Units 105 and 106 determine an optimum set of design parameters by limiting the sets of design parameters to corresponding sets of design parameters in the neighborhood of a Pareto boundary on the basis of the Pareto boundary of an objective function recognized from the usable areas displayed.

Abstract: An objective function can be mathematically approximated using a prescribed number of sample sets of design parameters and sets of a plurality of objective functions computed corresponding to them. A logical expression indicating a relation between or among arbitrary two or three objective functions of the plurality of mathematically approximated objective functions is computed as an inter-objective-function logical expression and a region that the arbitrary objective function values can take is displayed as a feasible region in an objective space corresponding to the arbitrary objective functions. Furthermore, a point or area in a design space corresponding to arbitrary design parameters corresponding to a point or area specified by a user in the displayed feasible region is displayed.

Abstract: A model parameter determination program makes a computer execute the procedures of substituting variable values and differential values of variables computed by using initial values of variables and model parameters for a first-order formula corresponding to the model; substituting supposition values for unknown model parameters within the post-substitution first-order formula; and applying a quantifier elimination method to the formula after the aforementioned substitution, thereby obtaining an upper limit value of errors.

Abstract: A program causes a computer to perform a step 1 of reading a model from a storage apparatus; a step 2 of computing difference equations by replacing variables with numeric values of a time-series that are stored in the storage apparatus; a step 3 of determining ranges of parameters of the model using a quantifier elimination method and a step 4 of verifying numbers of significant figures that show an accuracy of a computation by the determined ranges of parameters of the model.

Abstract: A model parameter determination program stored in a storage medium used to direct a computer to efficiently and effectively determine a parameter of a model in a simulation, and includes a step of receiving input of data of a variable in a model of a system to be analyzed; a step of generating a constraint expression relating to a parameter of a model based on the received data and the model of the system; and a step of obtaining presence/absence of a solution and/or a possible range of a feasible parameter by applying a quantifier elimination algorithm to the constraint expression.